JOHNSON'S 
TABLES 


"REESE  LIBRARY 
UNIVERSITY  OF  CALIFORNIA 


JOHNSON'S  TABLES. 


STADIA   AND    EARTH-WORK    TABLES 


FOUR-PLACE  LOGARITHMS,  LOGARITHMIC  TRAVERSE 

TABLE,   NATURAL   FUNCTIONS,  MAP 

PROJECTIONS,  ETC.,  ETC. 


REPRINTED   FROM 

THEORY  AND   PRACTICE  OF  SURVEYING. 


BY 

J.    B.    JOHNSON, 

PROFESSOR   OF   CIVIL    ENGINEERING,'  WASHINGTON    UNIVERSITY,   ST.    LOUIS 


NEW   YORK: 

JOHN    WILEY    &    SONS, 

53  EAST  TENTH  STREET. 

1892. 


COPYRIGHT,  1892, 

BY 

J.  B.  JOHNSON. 


7 


NOTE   BY  THE   AUTHOR. 


THE  great  use  made  by  engineers  of  three  of  the  following 
tables,  viz.,  the  Four-place  Logarithmic  Table,  the  Stadia 
Table,  and  the  table  giving  Prismoidal  Volumes,  has  necessi- 
tated the  binding  of  these  in  more  convenient  form  than  that 
in  which  they  first  appeared  in  the  Theory  and  Practice  of  Sur- 
veying. Since  the  cost  is  not  materially  increased  by  addi- 
tional pages,  the  remaining  tables  are  also  included,  as  well  as 
the  entire  chapter  on  the  Measurement  of  Volumes. 

The  Stadia  Tables  were  computed  by  Mr.  Arthur  Winslow, 
State  Geologist  of  Missouri,  and  first  published  by  the  Penn- 
sylvania Geological  Survey.  The  four-place  logarithm  tables 
were  originally  taken  from  Lee's  Tables  and  Formulae,  a  pub- 
lication of  the  U.  S.  Engineer  Corps.  The  table  giving  Vol- 
umes by  the  Prismoidal  Formula  was  computed  by  the  Author, 
It  is  the  only  table,  he  believes,  giving  volumes  by  the  pris- 
moidal  formula  at  one  operation.  It  may  also  be  used  for  Mean 
End-areas.  Tables  IV  and  VIII  are  also  original  in  their  ar- 
rangement. 

J.  B.  J. 
iii 


EXPLANATION   OF   TABLES. 


TABLES  I,  II,  III,  VI,  and  VII  require  no  explanation. 

TABLE  IV  gives  logarithmic  sines  and  cosines  to  four  places 
for  computing  latitudes  and  departures  when  the  angles  are 
read  from  zero  to  360  degrees.  It  can  of  course  be  used  for 
bearings  reading  from  zero  to  90  degrees,  as  is  ordinarily  done 
in  compass  work.  In  stadia  work,  and  always  in  transit  work 
where  the  instrument  is  graduated  continuously  to  360  degrees, 
this  table  will  be  found  very  convenient  for  coordinating  trav- 
erse lines,  as  well  as  for  computing  latitudes  and  departures  for 
closed  surveys. 

From  zero  to  5  degrees,  and  from  85  to  90  degrees,  the 
tables  give  values  for  each  minute  of  arc  without  tabular  dif- 
ferences. From  5  to  45  degrees  values  are  given  for  each  10 
minutes  of  arc  with  tabular  differences  for  the  log.  sines,  and 
from  45  to  85  degrees  with  tabular  differences  for  the  lo-minute 
increments  for  the  log.  cosines.  In  the  other  cases  the  tabular 
difference  is  so  small  as  to  be  readily  taken  at  sight.  Table 
IIIA  can  of  course  be  used  in  place  of  Table  IV  if  preferred. 

TABLE  V  gives  horizontal  distance  and  difference  of  elevation 
for  inclined  sights  in  stadia  work.  The  true  equations  of 
reduction  are  : 


Hor.  Dist.  —  r  cosaz>  +  (£+/)  cos  z/,  .     .     .     .     (i) 
and 

Dif.  Elev.  =  r  cos  v  sin  v  -f-  (c  -)-/)  sin  v  ;    .     .     (2) 


EXPLANATION  OF   TABLES. 


where 

r  —  reading  of  distance  on  stadia  rod  when  held  vertically ; 
v  =  vertical  angle  with  the  horizon  ; 
/=  focal  length  of  objective  ; 
c  =  distance  from  objective  to  centre  of  instrument. 

The  tables  give  the  values  for  the  first  term  only  of  the 
second  member.  The  values  for  the  second  term  are  given  at 
the  bottom  of  the  page,  the  constant  term  (c-\-f)  in  the  above 
equations  being  there  called  " c"  The  sum  of  these  two  dis- 
tances, viz.,  distance  from  centre  of  instrument  to  objective 
plus  distance  from  cross-wires  to  objective,  varies  in  different 
instruments  from  nine  to  fifteen  inches.  Three  values  of  this 
second  term  are  given,  therefore,  one  corresponding  to  c-\-f=. 
0.75  foot,  one  to  c-\-f-=>  i.oo  foot,  and  one  to  c-\-f=.  1.25 
foot.  In  ordinary  work  these  corrections  may  be  neglected. 
See  chapter  on  Stadia  Surveying  in  the  Theory  and  Practice 
of  Surveying. 

A  Reduction  Diagram,  printed  from  an  engraved  plate  20 
by  24  inches,  has  been  prepared  with  great  care,  giving  correc- 
tions to  the  horizontal  distance  read,  and  the  differences  of 
elevation,  for  inclined  sights,  as  shown  by  the  table,  not  includ- 
ing the  (£+/)  term.  For  all  angles  below  6°  and  distances 
less  than  1500  feet,  with  differences  of  elevation  less  than  50 
feet,  this  diagram  is  much  preferable  to  the  table.  The 
results  are  found  at  one  operation,  to  the  nearest  tenth  of  a 
foot,  with  great  rapidity.  It  can  be  procured  from  the  pub- 
lisher of  these  tables,  printed  on  heavy  lithographic  paper, 
price  50  cents,  post  paid. 

TABLE  VIII  gives  the  coordinates  to  be  used  in  the  poly- 
conic  projection  of  maps.  It  is  fully  explained  in  the  chapter 
on  Projection  of  Maps  in  the  Surveying. 

TABLES  IX  and  X  will  be  found  very  useful  in  sewer  and 
hydraulic  work  where  Kutter's  formula  is  to  be  used.  They 


Vi  EXPLANATION  OF  TABLES. 

are  fully  explained  in  the  chapter  on  Hydrographic  Survey- 
ing. 

TABLE  XI  gives  correct  volumes  of  prismoids,  by  the  pris- 
moidal  formula. 

For  the  benefit  of  railroad  engineers  and  others  who  either 
do  not  possess  a  copy  of  the  Surveying,  or  who  do  not  have  it 
by  them,  the  entire  chapter  on  the  Measurement  of  Volumes 
is  here  inserted.  At  least  seven  pages  of  this  chapter  is 
requisite  to  a  full  explanation  of  the  table,  and  for  the  sake  of 
completeness,  and  to  show  the  superiority  of  this  table  over 
any  table  of  volumes  from  mean  end-areas,  or  by  the  use  of 
diagonals,  it  has  been  thought  best  to  insert  the  entire  chap- 
ter. 

TABLE  XII  gives  the  azimuth  of  Polaris  at  any  hour-angle. 
By  its  use  an  observation  for  azimuth  to  the  nearest  minute  of 
arc  can  be  made  at  any  hour  when  the  star  is  visible,  provided 
the  local  time  is  known  to  within  one  or  two  minutes.  When 
the  observation  is  taken  two  hours  from  the  time  of  elongation, 
the  local  time  need  not  be  known  nearer  than  five  minutes. 
A  detailed  explanation  of  its  use  is  given  in  the  Surveying, 
Art.  38 1 A. 


CONTENTS. 


EXPLANATION  OF  TABLES iv 

THE  MEASUREMENT  OF  VOLUMES. 

310.  Proposition I 

311.  Grading  over  Extended  Surfaces 3 

312.  Approximate  Estimates  by  means  of  Contours 6 

313.  The  Prismoid II 

314.  The  Prismoidal  Formula II 

315.  Areas  of  Cross-sections 13 

316.  The  Centre  and  Side  Heights 14 

317.  The  Area  of  a  Three-level  Section 14 

318.  Cross-sectioning , 15 

319.  Three-level  Sections,  the  Upper  Surface  consisting  of  two  Warped 

Surfaces 17 

320.  Construction  of  Tables  for  Prismoidal  Computation 19 

321.  Three-level  Sections,  the  Upper  Surface  divided  into  Four  Planes 

by  Diagonals 24 

322.  Comparison  of  Volumes  by  Diagonals  and  by  Warped  Surfaces. .  26 

323.  Preliminary  Estimates  from  the  Profiles 28 

324.  Borrow  Pits 31 

325.  Shrinkage  of  Earthwork  31 

326.  Excavations  under  Water 32 

TABLES. 

I.  TRIGONOMETRICAL  FORMULAE 37 

II.  FOR  CONVERTING  METERS,  FEET,  AND  CHAINS 41 

III.  LOGARITHMS  OF  NUMBERS  TO  FOUR  PLACES 42 

IIlA.  LOGARITHMS  OF  TRIGONOMETRICAL  FUNCTIONS  TO  FOUR  PLACES..  44 

IV.  LOGARITHMIC  TRAVERSE  TABLE 48 

V.  STADIA  REDUCTIONS  FOR  HORIZONTAL  DISTANCE  AND  FOR  ELEVA- 
TION   56 

VI.  NATURAL  SINES  AND  COSINES 64 

VII.  NATURAL  TANGENTS  AND  COTANGENTS 73 

VIII.  COORDINATES  FOR  POLYCONIC  PROJECTION 85 

IX.  VALUES  OF  COEFFICIENTS  IN  KUTTER'S  FORMULA 86 

X.  DIAMETERS  OF  CIRCULAR  CONDUITS  BY  KUTTER'S  FORMULA 87 

XI.  EARTHWORK  TABLE — VOLUMES  BY  THE  PRISIMOIDAL  FORMULA.  ...  88 

XII.  AZIMUTHS  OF  POLARIS  AT  ALL  HOUR  ANGLES 98 

vii 


CHAPTER    XIII. 
THE   MEASUREMENT   OF   VOLUMES. 

310.  Proposition. —  The  volume  of  any  doubly-truncated 
prism  or  cylinder,  bounded  by  plane  ends,  is  equal  to  the  area  of  a 
right  section  into  the  length  of  the  element  through  the  centres  of 
gravity  of  the  bases,  or  it  is  equal  to  the  area  of  either  base  into 
the  altitude  of  the  element  joining  the  centres  of  gravity  of  the 
bases,  measured  perpendicular  to  that  base. 

Let  ABCD,  Fig.  107,  be  a  cylinder,  cut  by  the  planes  OC 
and  OB,  the  unsymmetrical  right  section  EF  being  shown  in 
plan  in  E'F'.  Whatever  position  the  cutting  planes  may  have, 
if  they  are  not  parallel  they  will  intersect  in  a  line.  This  line 
of  intersection  may  be  taken  perpendicular  to  the  paper,  and 
the  body  would  then  appear  as  shown  in  the  figure,  the  line 
of  intersection  of  the  cutting  planes  being  projected  at  O. 

Let  A  =  area  of  the  right  section ; 

A A  =  any  very  small  portion  of  this  area; 

x  =  distance  of  any  element  from  O ; 
then  ax  =  height  of  any  element  at  a  distance  x  from  <9. 

An  elementary  volume  would  then  be  ax  A  A,  and  the  total 
volume  of  the  solid  would  be  "Sax  A  A. 

Again,  the  total  volume  is  equal  to  the  mean  or  average 
height  of  all  the  elementary  volumes  multiplied  by  the  area 
of  the  right  section. 

The  mean  height  of  the  elementary  volumes  is,  therefore, 


OF  THF 

UNIVERSITY 


SURVEYING. 


But 


A 


is  the  distance  from  O  to  the 


centre  of  gravity,  G,  of  the  right  section,*  and  a  times  this  dis« 
tance  is  the  height  of  the  element  LK  through  this  point. 
Therefore,  the  mean  height  is  the  height  through  the  centre  of 


•  PIG.  107. 

gravity  of  the  base,  and  this  into  the  area  of  the  right  section 
is  the  volume  of  the  truncated  prism  or  cylinder.  The  truth 
of  the  alternative  proposition  can  now  readily  be  shown. 

Corollary.  When  the  cylinder  or  prism  has  a  symmetrical 
cross-section,  the  centre  of  gravity  of  the  base  is  at  the  centre 
of  the  figure,  and  the  length  of  the  line  joining  these  centres 
is  the  mean  of  any  number  of  symmetrically  chosen  exterior 
elements.  For  instance,  if  the  right  section  of  the  prism  be  a 
regular  polygon,  the  height  of  the  centre  element  is  the  mean 
of  the  length  of  all  the  edges.  This  also  holds  true  for  paral- 
lelograms, and  hence  for  rectangles.  Here  the  centres  of  gravity 

*  This  is  shown  in  mechanics,  and  the  student  may  have  to  take  it  for 
granted  temporarily. 


THE  MEASUREMENT  OF    VOLUMES. 


of  the  bases  lie  at  the  intersections  of  the  diagonals ;  and  since 
these  bisect  each  other,  the  length  of  the  line  joining  the  in- 
tersections is  the  mean  of  the  lengths  of  the  four  edges.  The 
same  is  true  of  triangular  cross-sections. 

311.  Grading  over  Extended  Surfaces. — Lay  out  the 
area  in  equal  rectangles  of  such  a  size  that  the  surfaces  of  the 
several  rectangles  may  be  considered  planes.  For  common 
rolling  ground  these  rectangles  should  not  be  over  fifty  feet 
on  a  side.  Let  Fig.  108  represent  such  an  area.  Drive  pegs  at 


4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

a        2        i 

123             3            1 

FIG.  108. 

the  corners,  and  find  the  elevation  of  the  ground  at  each  in- 
tersection by  means  of  a  level,  reading  to  the  nearest  tenth  of 
a  foot,  and  referring  the  elevations  to  some  datum-plane  below 
the  surface  after  it  is  graded.  When  the  grading  is  completed, 
relocate  the  intersections  from  witness-points  that  were  placed 
outside  the  limits  of  grading,  and  again  find  the  elevations  at 
these  points.  The  several  differences  are  the  depths  of  excava- 
tion (or  fill)  at  the  corresponding  corners.  The  contents  of 
any  partial  volume  is  the  mean  of  the  four  corner  heights  into 
the  area  of  its  cross-section.  But  since  the  rectangular  areas 
were  made  equal,  and  since  each  corner  height  will  be  used  as 
many  times  as  there  are  rectangles  joining  at  that  corner,  we 
have,  in  cubic  yards, 


r= 


4x27 


SUX  VE  YING. 


The  subscripts  denote  the  number  of  adjoining  rectangles 
the  area  of  each  of  which  is  A. 

From  this  equation  we  may  frame  a 

RULE. — Take  each  corner  height  as  many  times  as  there 
are  partial  areas  adjoining  it,  add  them  all  together,  and  mul- 
tiply by  one  fourth  of  the  area  of  a  single  rectangle.  Tnis 
gives  the  volume  in  cubic  feet.  To  obtain  it  in  cubic  yards, 
divide  by  twenty-seven. 

If  the  ground  be  laid  out  in  rectangles,  30  feet  by  36  feet, 

A  1080 

then  — — — •  =  — ~-  =  10 ;  and  if  the  elevations  be  taken  to 

the  nearest  tenth  of  a  foot,  then  the  sum  of  the  multiplied 
corner  heights,  with  the  decimal  point  omitted,  is  at  once  the 
the  amount  of  earthwork  in  cubic  yards.  This  is  a  common 
way  of  doing  this  work.  In  borrow-pits,  for  which  this  method 
is  peculiarly  fitted,  the  elementary  areas  would  usually  be 
smaller. 

In  general,  on  rolling  ground,  a  plane  cannot  be  passed 
through  the  four  corner  heights.  We  may,  however,  pass  a 
plane  through  any  three  points,  and  so  with  four  given  points 


FIG.  109. 


on  a  surface  either  diagonal  may  be  drawn,  which  with  the 
bounding  lines  makes  two  surfaces.  If  the  ground  is  quite 
irregular,  or  if  the  rectangles  are  taken  pretty  large,  the  sur- 
veyor may  note  on  the  ground  which  diagonal  would  most 


THE  MEASUREMENT  OF    VOLUMES.  5 

nearly  fit  the  surface.  Let  these  be  sketched  in  as  shown  in 
Fig.  109.  Each  rectangular  area  then  becomes  two  triangles, 
and  when  computed  as  triangular  prisms,  each  corner  height 
at  the  end  of  a  diagonal  is  used  twice,  while  the  two  other 
corner  heights  are  used  but  once.  That  is,  twice  as  much 
weight  is  given  to  the  corner  heights  on  the  diagonals  as  to 
the  others.  In  Fig.  109,  the  same  area  as  that  in  Fig.  108  is 
shown  with  the  diagonals  drawn  which  best  fit 
the  surface  of  the  ground.  The  numbers  at 
the  corners  indicate  how  many  times  each 
height  is  to  be  used.  It  will  be  seen  that 
each  height  is  used  as  many  times  as  there  are 
triangles  meeting  at  that  corner.  To  derive 
the  formula  for  this  case,  take  a  single  rectangle,  as  in  Fig. 
no,  with  the  diagonal  joining  corners  2  and  4.  Let  A  be  the 
area  of  the  rectangle.  Then  from  the  corollary,  p.  395,  we 
have  for  the  volume  of  the  rectangular  prism,  in  cubic  yards, 


rr _    "          *  '"\      I     r"2     I     '"*     [     '"a 

M     4*      V/'     S%*V    1  /> 


2X2;\          3  3 

A 
6  X  27 


-  (k,  +  2h,  +  ^  +  2/0 (2) 


For  an  assemblage  of  such  rectangular  prisms  as  shown  in 
Fig.  109,  the  diagonals  being  drawn,  we  have,  in  cubic  yards, 


;    .     .     .     (3) 

where  A  is  the  area  of  one  rectangle,  and  the  subscripts  denote 
the  number  of  triangles  meeting  at  a  corner, 


SURVEYING. 


As  a  check  on  the  numbering  of  the  corners,  Fig.  109,  add 
them  all  together  and  divide  by  six.  The  result  should  be 
the  number  of  rectangles  in  the  figure.  In  this  case,  if  the 
rectangles  be  taken  36  feet  by  45  feet,  or,  better,  40  feet  by  40.5. 
feet,  then  the  sum  of  the  multiplied  heights  with  the  decimal 
point  omitted  is  the  number  of  cubic  yards  of  earthwork,  the 
corner  heights  having  been  taken  out  to  tenths  of  a  foot. 

The  method  by  diagonals  is  more  accurate  than  that  by 
rectangles  simply,  the  dimensions  being  the  same ;  or,  for 
equal  degrees  of  exactness  larger  rectangles  may  be  used  with 
diagonals  than  without  them,  and  hence  the  work  materially 
reduced.  In  any  case  some  degree  of  approximation  is  neces- 
sary. 

312.  Approximate  Estimates  by  means  of  Contours. — 
(A)  Whenever  an  extended  surface  of  irregular  outline  is  to 
be  graded  down,  or  filled  up  to  a  given  plane  (not  a  warped  or 
curved  surface),  a  near  approximation  to  the  amount  of  cut  or 
fill  may  be  made  from  the  contour  lines.  In  Fig.  1 1 1  the  full 
curved  lines  are  contours,  showing  the  original  surface  of  the 
ground.  Every  fifth  one  is  numbered,  and  these  were  the  con- 
tours shown  on  the  original  plat.  Intermediate  contours  one 
foot  apart  have  been  interpolated  for  the  purpose  of  making 
this  estimate.  The  figures  around  the  outside  of  the  bound- 
ing lines  give  the  elevations  of  those  points  after  it  is  graded 
down.  The  straight  lines  join  points  of  equal  elevation  after 
grading ;  and  since  this  surface  is  to  be  a  plane  these  lines  are 
surface  or  contour  lines  after  grading.  Wherever  these  two 
sets  of  contour  lines  intersect,  the  difference  of  their  elevations 
is  the  depth  of  cut  or  fill  at  that  point.  If  now  we  join  the 
points  of  equal  cut  or  fill  (in  this  case  it  is  all  in  cut),  we  ob- 
tain a  new  set  of  curves,  shown  in  the  figure  by  dotted  lines, 
which  may  be  used  for  estimating  the  amount  of  earthwork. 
The  dotted  boundaries  are  the  horizontal  projections  of  the 
traces  on  the  natural  surface  of  planes  parallel  to  the  final 


THE  MEASUREMENT  OF   VOLUMES. 


graded  surface  which  are  uniformly  spaced  one  foot  apart  ver- 
tically. These  projected  areas  are  measured  by  the  planimeter 
and  called  Al}  A2,  As,  etc.  Each  area  is  bounded  by  the 
dotted  line  and  the  bounding  lines  of  the  figure,  since  on  these 


78 

FIG.  in. 


bounding  lines  all  the  projections  of  all  the  traces  unite,  the 
slope  here  being  vertical.  For  any  two  adjoining  layers  we 
have,  by  the  prismoidal  formula*  as  well  as  by  Simpson's  one- 
third  rule, 


(I) 


where   h   is  the  common  vertical  distance  between  the  pro- 
jected areas. 

*  For  the  demonstration  of  the  prismoidal  formula  see  Art.  314. 


SURVEYING. 


For  the  next  two  layers  we  would  have,  similarly, 

r,-i=|<^.-K4A^»);.  .".-  .....  (2) 

or  for  any  even  number   of   layers  we  would  have,  in    cubic 
yards, 

V=^  +  *A*  +  2A,  +  4At  +  2Af  +    ....  A.\  (3) 


where  n  is  an  odd  number,  h  and  A  being  in  feet  and  square 
feet  respectively. 

(B)  Whenever  the  final  surface  is  not  to  be  a  plane,  but 
warped,  undulating,  or  built  to  regular  outlines  like  a  fortifi- 
cation, a  reservoir  embankment,  or  terraced  grounds,  a  differ- 
ent method  should  be  employed. 

In  the  former  method  the  areas  bounded  by  the  dotted 
lines  were  areas  cut  out  by  planes  parallel  to  the  final  plane 
surface,  passed  one  foot  apart  vertically.  But  since  the  map 
shows  only  the  horizontal  projections  of  these  planes,  these  pro- 
jections, multiplied  by  the  vertical  distance  between  them, 
would  give  the  true  volumes. 

When  the  final  surface  is  not  to  be  a  plane,  proceed  as  fol- 
lows :  First  make  a  careful  contour  map  of  the  ground.  Then 
lay  down  on  this  map  a  system  of  contour  lines,  corresponding 
in  elevation  to  the  first  set  of  contours,  but  in  a  different 
colored  ink,  which  will  accurately  represent  the  final  surface 
desired.  This  second  set  of  contours  would  be  a  series  of 
straight  lines  if  a  regular  surface,  composed  of  plane  faces,  was 
to  be  constructed,  but  would  be  curving  lines  if  the  ground 
were  to  be  brought  to  a  final  curving  or  undulating  surface. 

The  closed  figures  bounded  by  the  two  sets  of  intersecting 
contours  of  the  same  elevation  are  horizontal  areas  of  cut 
or  fill,  separated  by  the  common  vertical  distance  between 


THE  MEASUREMENT  OF   VOLUMES. 


contours.  The  volumes  here  defined  are  oblique  solids 
bounded  by  horizontal  planes  at  top  and  bottom,  and  are  a 
species  of  prismoid.  The  volume  of  one  of  these  prismoids  is 
found  by  applying  the  prismoidal  formula  to  it,  finding  the  end 
areas  by  means  of  a  planimeter,  and  taking  the  length  as  the 


660 
Fig.   Ilia. 


vertical  distance  between  contours.  If  the  contours  be  drawn 
close  enough  together,  then  each  alternate  contour-area  may  be 
used  as  a  middle  area,  and  the  length  of  the  prismoid  taken  at 
twice  the  vertical  distance  between  contours ;  or  the  volume 


10  SURVEYING. 


may  be  computed  by  either  of  the  formulas  (12),  (13),  (14),  or 
(15)  of  Appendix  C,  where  the  /is  would  here  become  the  end 
areas  and  /  the  vertical  distance  between  contours. 

Example :  Let  it  be  required  to  build  a  square  reservoir  on 
a  hillside,  which  shall  be  partly  in  excavation  and  partly  in* 
embankment,  the  ground  being  such  as  shown  by  the  full  con- 
tour lines  in  Fig.  iii#.* 

The  contours,  for  the  sake  of  simplicity  and  brevity,  are 
spaced  five  feet  apart.  The  top  of  the  wall,  shown  by  the  full 
lines  making  the  square,  is  10  feet  wide  and  at  an  elevation  of 
660  feet.  The  reservoir  is  20  feet  deep,  with  side  slopes,  both 
inside  and  outside,  of  two  to  one,  making  the  bottom  elevation 
640  feet,  and  20  feet  square,  the  top  being  ico  feet  square  on 
the  inside.  The  dotted  lines  are  contours  of  the  finished 
slopes,  both  inside  and  out,  at  elevations  shown  on  the  figure. 
The  areas  in  fill  all  fall  within  the  broken  line  marked  a  b  c  d  e 
f  g  h  i  k,  and  the  cut  areas  all  fall  within  the  broken  line 
marked  a  b  c  def  g  o.  These  broken  lines  are  grade  lines. 
The  horizontal  sectional  areas  in  fill  and  cut  are  readily  traced 
by  following  the  closed  figures  formed  by  contours  of  equal 
elevation,  thus — 

At  640  foot  level  sectional  area  in  fill  is/  s  t. 
"    650     "       "  "  "         "         Imn  u  v  x  I. 

"    650     "       "  "  "          cut  is  i  2  3  u  x. 

The  other  areas  are  as  easily  traced.  In  the  figure  the  lines 
have  all  been  drawn  in  black.  In  practice  they  should  be 
drawn  in  different  colors  to  avoid  confusion. 

This  second  method  should  be  used  in  all  cases  where  the 
graded  area  is  considerable  and  the  final  relief  form  is  not  a 
plane.  If  the  contours  be  carefully  determined  and  be  taken 

*  This  figure  is  taken  from  a  paper  describing  the  method  by  Prof.  William 
G.  Raymond,  University  of  California. 


THE  MEASUREMENT  OF    VOLUMES.  \\ 

near  enough  together,  the  method  will  give  as  accurate  results 
as  may  be  obtained  in  any  other  way.  The  volume  may  be 
computed  by  eq.  (3)  of  this  article,  where  the  areas  are  the 
horizontal  sectional  areas  bounded  by  contours  of  equal  ele- 
vation, and  h  is  the  vertical  distance  between  contours. 

When  these  methods  are  used  for  final  estimates,  the  con- 
tours should  be  carefully  determined,  and  spaced  not  more 
than  two  feet  apart  on  steep  slopes  and  one  foot  apart  on  low 
slopes. 

313.  The   Prismoid  is  a  solid  having  parallel  end  areas, 
and  may  be  composed  of  any  combination  of  prisms,  cylinders, 
wedges,  pyramids,  or  cones  or  frustums  of  the  same,  whose 
bases  and  apices  lie  in  the  end  areas.     It  may  otherwise  be 
defined  as  a  volume  generated  by  a  right-line  generatrix  mov- 
ing on  the  bounding  lines  of  two  closed  figures  of  any  shapes 
which  lie  in  parallel  planes  as  directrices,  the  generatrix  not 
necessarily  moving  parallel  to  a  plane  director.     Such  a  solid 
would   usually  be   bounded  by  a  warped    surface,  but  it    can 
always  be  subdivided  into  one  or  more  of   the  simple  solids 
named  above. 

Inasmuch  as  cylinders  and  cones  are  but  special  forms  of 
prisms  and  pyramids,  and  warped  surface  solids  may  be  divided 
into  elementary  forms,  of  them,  and  since  frustums  may  also 
be  subdivided  into  the  elementary  forms,  it  is  sufficient  to  say 
that  all  prLmoids  may  be  decomposed  into  prisms,  wedges, 
and  pyramids.  If  a  formula  can  be  found  which  is  equally 
applicable  to  all  of  these  forms,  then  it  will  apply  to  any  com- 
bination of  them.  Such  a  formula  is  called 

314.  The  Prismoidal  Formula. 

Let  A  =  area  of  the  base  of  a  prism,  wedge,  or  pyramid  ; 
A^  Am,  A9  =  the  end  and  middle  areas  of  a  prismoid,  or  of  any 

of  its  elementary  solids ; 
h  —  altitude  of  the  prismoid  or  elementary  solid. 


12  SURVEYING. 


Then  we  have, 
For  Prisms, 

V=  hA  =  g-  (Al  +  4^TO  +  ^3) (i) 

For  Wedges, 

2         6^1'        *  ^ 
For  Pyramids, 


(3) 


Whence  for  any  combination  of  these,  having  all  the  common 
altitude  h,  we  have 


(4) 


which  is  the  prismoidal  formula. 

It  will  be  noted  that  this  is  a  rigid  formula  for  all  prismoids. 
The  only  approximation  involved  in  its  use  is  in  the  assump- 
tion that  the  given  solid  may  be  generated  by  a  right  line 
moving  over  the  boundaries  of  the  end  areas. 

This  formula  is  used  for  computing  earthwork  in  cuts  and 
fills  for  railroads,  streets,  highways,  canals,  ditches,  trenches, 
levees,  etc.  In  all  such  cases,  the  shape  of  the  figure  above 
the  natural  surface  in  the  case  of  a  fill,  or  below  the  natural 
surface  in  the  case  of  a  cut,  is  previously  fixed  upon,  and  to 
complete  the  closed  figure  of  the  several  cross-section  areas 
only  the  outline  of  the  natural  surface  of  the  ground  at  the 
section  remains  to  be  found.  These  sections  should  be  located 
so  near  together  that  the  intervening  solid  may  fairly  be  as- 


THE  MEASUREMENT  OF   VOLUMES. 


sumed  to  be  a  prismoid.  They  are  usually  spaced  100  feet 
apart,  and  then  intermediate  sections  taken  if  the  irregularities 
seem  to  require  it. 

The  area  of  the  middle  section  is  never  the  mean  of  the 
two  end  areas  if  the  prismoid  contains  any  pyramids  or  cones 
among  its  elementary  forms.  When  the  three  sections  are 
similar  in  form,  the  dimensions  of  the  middle  area  are  always 
the  means  of  the  corresponding  end  dimensions.  This  fact 
often  enables  the  dimensions,  and  hence  the  area  of  the  middle 
section,  to  be  computed  from  the  end  areas.  Where  this  can- 
not be  done,  the  middle  section  must  be  measured  on  the 
ground,  or  else  each  alternate  section,  where  they  are  equally 
spaced,  is  taken  as  a  middle  section,  and  the  length  of  the 
prismoid  taken  as  twice  the  distance  between  cross-sections. 
For  a  continuous  line  of  earthwork,  we  would  then  have,  in 
cubic  yards, 


0.    •     (0 


where  /  is  the  distance  between  sections  in  feet.  This  is  the 
same  as  equation  (3),  p.  401.  Here  the  assumption  is  made 
that  the  volume  lying  between  alternate  sections  conforms 
sufficiently  near  to  the  prismoidal  forms. 

315.  Areas  of  Cross-sections.  — Inmost  cases,  in  practice 
at  least,  three  sides  of  a  cross-section  are  fixed  by  the  conditions 
of  the  problem.  These  are  the  side  slopes  in  both  cuts  and 
fills,  the  bottom  in  cuts  and  the  top  in  embankments,  or  fills. 
It  then  remains  simply  to  find  where  the  side  slopes  will  cut 
the  natural  surface,  and  also  the  form  of  the  surface  line  on  the 
given  section.  Inasmuch  as  stakes  are  usually  set  at  the  points 
where  the  side  slopes  cut  the  surface,  whether  in  cut  or  fill, 
such  stakes  are  called  slope-stakes,  and  they  are  set  at  the  time 


14  SURVEYING. 


the  cross-section  is  taken.  The  side  slopes  are  defined  as  so 
much  horizontal  to  one  vertical.  Thus  a  slope  of  i£  to  I  means 
that  the  horizontal  component  of  a  given  portion  of  a  slope- 
line  is  it^  times  its  vertical  component,  the  horizontal  com- 
ponent always  being  named  first.  The  slope-ratio  is  the  ratio 
of  the  horizontal  to  the  vertical  component,  and  is  therefore 
always  the  same  as  the  first  number  in  the  slope-definition. 
Thus  for  a  slope  of  i£  to  I  the  slope-ratio  is  I J. 

316.  The  Centre  and  Side  Heights.— The  centre  heights 
are  found  from  the  profile  of  the  surface  along  the  centre  line, 
on  which  has  been  drawn  the  grade  line  of  the  proposed  work. 
These  are  carefully  drawn  on  cross-section    paper,  when  the 
height  of  grade  at  each  station  above  or  below  the  surface  line 
can  be   taken  off.      These  centre  heights,  together  with  the 
width  of  base  and  side  slopes  in  cuts  and  in  fills,  are  the  neces- 
sary data  for  fixing  the  position  of  the  slope-stakes.     When 
these  are  set  for  any  section  as  many  points  on  the  surface 
line  joining  them  may  be  taken  as  desired.     In  ordinary  rolling 
ground  usually  no  intermediate  points  are  taken,  the  centre 
point  being  already  determined.     In  this  case  three  points  in 
the  surface  line  are  known,  both  as  to  their  distance  out  from 
the  centre  line  and  as  to  their  height  above  the  grade  line. 
Such  sections  are  called  "  three-level  sections,"  the  surface  lines 
being   assumed  straight  from  the  slope-stakes   to  the  centre 
stake. 

317.  The  Area  of  a  Three-level  Section. 
Let  d  and  df  be  the  distances  out,  and 

h  and  ti  the  heights  above  grade  of  right  and  left  slope- 
stakes,  respectively; 
D  the  sum  ot  d  and  d', 
c    the  centre  height, 
r    the  slope-ratio, 
w    the  width  of  bed. 


THE  MEASUREMENT  OF    VOLUMES. 


Then  the  area  ABCDE  is  equal  to  the  sum  of  the  four  trian- 
gles A£w,  BCwt  wCD,  and  wED.     Or, 


w 


A  = 


(0 


This  area  is  also  equal  to  the  sum  of  the  triangles  FCD  and 
FED,  minus  the  triangle  AFB.     Or, 


D      it? 


FIG. 


Equation  (2)  can  also  be  obtained  directly  from  equation 

(1)  by  substituting  for  h  and  h'  in  (i)  their  values  in  terms  of 

<*--* 

d  and  w,  h  =  —    — ,  and  then  putting  D  =  d-\-  d' .    Equation 

(2)  has  but  two  variables,  c  and  D,  and  is  the  most  convenient 
one  to  use. 

318.  Cross-sectioning. — It  will  be  seen  from  Fig.  112  that 
in  the  case  of  a  three-level  section  the  only  quantities  to  be 
determined  in  the  field  are  the  heights,  h  and  k ',  and  the  dis- 
tances out,  d  and  d' ,  of  the  slope-stakes.  These  are  found  by 
trial.  A  levelling  instrument  is  set  up  so  as  to  read  on  the 


SURVEYING. 


three  points  C,  D^  £,  and  the  rod  held  first  atZ>.  The  reading 
here  gives  the  height  of  instrument  above  this  point.  Add 
this  algebraically  to  the  centre  height  (which  may  be  negative, 
and  which  has  been  obtained  from  the  profile  for  each  station), 
and  the  sum  is  the  height  of  instrument  above  (or  below)  the 
grade  line.  If  the  ground  were  level  transversely,  the  distance 
out  to  the  slope-stakes  would  be 


w 

-. 


But  this  is  not  usually  the  case,  and  hence  the  distance  out 
must  be  found  by  trial.  If  the  ground  slopes  j  °wn  1 

from  the  centre  line  in  a  j      ,  >•  the  distance  out  will  evidently 

be  more  than  that  given  by  the  above  equation,  and  vice  versa. 
The  rodman  estimates  this  distance,  and  holds  his  rod  at  a  cer- 
tain measured  distance  out,  d^  The  observer  reads  the  rod, 
and  deducts  the  reading  from  the  height  of  instrument  above 
grade  (or  adds  it  to  the  depth  of  instrument  below  grade),  and 
this  gives  the  height  of  that  point,  hlt  above  or  below  grade.  Its 

IV 

distance  out,  then,  should  bed  =  hf  -\ — .     If  this  be  more  than 

2 

the  actual  distance  out,  dl9  the  rod  is  set  farther  out ;  if  less,  it 
is  moved  in.  The  whole  operation  is  a  very  simple  one  in  prac- 
tice, and  the  rodman  soon  becomes  very  expert  in  estimating 
nearly  the  proper  position  the  first  time. 

In  heavy  work — that  is,  for  large  cuts  or  fills,  and  for  irregu- 
lar ground — it  may  be  necessary  to  take  the  elevation  and  dis- 
tance out  of  other  points  on  the  section  in  order  to  better 
determine  its  area.  These  are  taken  by  simply  reading  on  the 
rod  at  the  critical  points  in  the  outline,  and  measuring  the  dis- 
tances out  from  the  centre.  The  points  can  then  be  plotted 


THE  MEASUREMENT  OF    VOLUMES. 


on  cross-section  paper  and  joined  by  straight  or  by  free-hand 
curved  lines.  In  the  latter  case  the  area  should  be  deter- 
mined by  planimeter. 

319.  Three-level  Sections,  the  Upper  Surface  con- 
sisting of  two  Warped  Surfaces.  —  If  the  three  longitudinal 
lines  joining  the  centre  and  side  heights  on  two  adjacent  three- 
level  sections  be  used  as  directrices,  and  two  generatrices,  one 
on  each  side  the  centre,  be  moved  parallel  to  the  end  areas  as 
plane  directers,  two  warped  surfaces  are  generated,  every  cross- 
section  of  which  parallel  to  the  end  areas  is  a  three-level  sec- 
tion. These  same  surfaces  could  be  generated  by  two  longi- 
tudinal generatrices,  moving  over  the  surface  end-area  lines  as 
directrices.  The  surface  would  therefore  be  a  prismoid,  and 
its  exact  volume  would  be  given  by  the  prismoidal  formula. 
The  middle  area  in  this  case  is  readily  found,  since  the  center 
and  side  heights  are  the  means  of  the  corresponding  end  di- 
mensions. 

The  prismoidal  formula,  giving  volumes  in  cubic  yards, 


could  therefore  be  written 


This  equation  is  derived  directly  from  eq.  (i)  above,  and  eq. 


(2),  p.  406.   The  quantity  —  is  the  distance  from  the  grade-plane 


1 8  SURVEYING. 


to  the  intersection  of  the  side  slopes,  and  is  a  constant  for  any 
given  piece  of  road.     It  would  have  different  values,  however, 
in  cuts  and  fills  on  the  same  line. 
For  brevity,  let 


—  =  c  -         and  w       =       °   -  K 

zr  4  X  27?-     '    54 

Here  K  is  the  volume  of  the  prism  of  earth,  100  feet  long,  in- 
cluded between  the  roadbed  and  side  slopes.     It  is  first  in- 
cluded in  the  computation  and  then  deducted.     It  is  also  a 
constant  for  a  given  piece  of  road. 
Equation  (2)  now  becomes 


-#;  .  (3) 


where  cm  and  Dm  are  the  means  of  c^t  and  DJ)V  respectively. 
This  equation  involves  but  two  kinds  of  variables,  c  and  D, 
and  is  well  adapted  to  arithmetical,  tabular,  or  graphical  com- 
putation. Thus  if  /  =  100  ;  w  —  18  ;  and  r  =  i-J-  ;  then  c0  —  6  ; 
and  K  =  200  ;  and  equation  (3)  becomes 


',  +  6)  A  +  ('.  +  6)A  +  4fc.  +  6)  AJ  -  2oo  •  (4) 

If  the  total  centre  heights  (to  intersection  of  side  slopes)  be 
represented  by  Clt  Cv  and  Cm,  then  eq.  (3)  becomes,  in  general, 

V=lC(ClDl  +  CJ)%  +  4CJ)m)-Kt.    .    .    (5) 

where  K'  =  -J^j-,  and  is  independent  of  width  of  bed  and  of 
slopes. 

For  any  given  piece  of  road,  the  constants  K,  K'  ,  and  c0  are 
known,  and  for  each  prismoid  the  C's  and  D's  are  observed, 
hence  for  any  prismoid  all  the  quantities  in  eq.  (5)  are  known. 


THE  MEASUREMENT  OF    VOLUMES. 


320.  Construction  of  Tables  for  Prismoidal  Computa- 
tion. —  If  a  table  were  prepared  giving  the  products  K'CD  for 
various  values  of  C  and  D,  it  could  be  used  for  evaluating 
equation  (3),  which  is  the  same  as  equation  (5).  The  argu- 
ments would  be  the  total  widths  (Z^),  and  the  centre  heights 
(£).  Such  a  table  would  have  to  be  entered  three  times  for 
each  prismoid,  first  with  C,  and  Dl  ;  second  with  Ct  and  Z>a  ; 
and  finally  with  Cm  and  Dm.  If  four  times  the  last  tabular 
value  be  added  to  the  sum  of  the  other  two,  and  K  subtracted, 
the  result  is  the  true  volume  of  the  prismoid. 


VALUES  OF   c 


(=  -) 

\        2rl 


AND 


4  X 

AND  SLOPES. 


-     FOR  VARIOUS  WIDTHS 


Width 
of 
Road- 

SCOPES. 

X  to  1. 

Ntol. 

%  to  1. 

1  to  1. 

Ik  tol. 

IN  tol. 

IX  tol. 

2  to  1. 

bed. 

Co 

JC 

c; 

jr 

C0 

K 

C0 

K 

c« 

K 

Co 

K 

Co 

K 

C0 

K 

1O 

20 

370 

10 

185 

6.7 

123 

5-o 

93 

4.0 

74 

3-3 

62 

2.9 

53 

2.5 

46 

11 

22 

448 

ii 

224 

7-3 

149 

5-5 

112 

4.4 

9° 

3-7 

75 

3-i 

64 

2.8 

56 

13 

24 

533 

12 

266 

8.0 

I78 

6.0 

J33 

4.8 

107 

4.0 

89 

3-4 

76 

3-o 

67 

13 

26 

626 

13 

313 

8.7 

209 

6-5 

157 

5-2 

125 

4-3 

104 

3-7 

89 

3-2 

78 

14 

28 

725 

M 

363 

9-3 

242 

7.0 

z8z 

5-6 

*45 

4-7 

121 

4.0 

104 

3-5 

9* 

15 

3° 

833 

15 

4T7 

IO.O 

278 

7-5 

208 

6.0 

167 

S-o 

139 

4-3 

119 

3-8 

104 

16 

32 

948 

16 

474 

10.7 

3,6 

8.0 

237 

6.4 

190 

5-3 

158 

4.6 

i35 

4.0 

118 

17 

34 

1070 

77 

535 

"•3 

357 

8-5 

268 

6.8 

214 

5-7 

I78 

4-9 

i53 

4-2 

134 

18 

36 

1200 

18 

600 

12.0 

400 

9.0 

300 

7.2 

240 

6.0 

200 

S-i 

171 

4-5 

I5<> 

19 

38 

'337 

J9 

668 

I2.7 

446 

9-5 

334 

7.6 

267 

6-3 

223 

4-4 

191 

4.8 

167 

30 

40 

1481 

20 

740 

*3-3 

494 

IO.O 

370 

8.0 

296 

6-7 

247 

5-7 

212 

5.0 

185 

31 

42 

1633 

21 

816 

14.0 

544 

IO-5 

408 

8.4 

327 

7.0 

272 

6.0 

233 

5-2 

204 

33 

44 

1793 

22 

896 

14.7 

598 

ii  .0 

448 

8.8 

359 

7-3 

299 

6.3 

256 

5-5 

224 

33 

46 

J959 

23 

980 

iS-3 

653 

"•5 

490 

9.2 

392 

7-7 

326 

6.6 

280 

5-8 

245 

34 

48 

2i34 

24 

1067 

16.0 

711 

12.  0 

534 

9.6 

427 

8.0 

356 

6.9 

305 

6.0 

267 

35 

5° 

2315 

25 

1158 

16.7 

772 

12-5 

579 

IO.O 

463 

8-3 

386 

7-i 

331 

6.2 

264 

36 

S* 

2504 

26 

1252 

17-3 

835 

13.0 

626 

10.4 

50i 

8-7 

417 

7-4 

358 

6-5 

3^3 

37 

54 

2700 

27 

i35o 

18.0 

900 

13-5 

675 

10.8 

54° 

9.0 

450 

7-7 

386 

6.8 

338 

38 

56 

2904 

28 

*452 

18.7 

968 

14.0 

726 

IZ.2 

581 

93 

484 

8.0 

4i5 

7.0 

363 

39 

58 

3"5 

29 

1558 

i9-3 

1038 

14.5 

779 

ii.  6 

623 

9-7 

519 

8  3 

445 

7-2 

389 

3O 

60 

3333 

3° 

1667 

20.0 

IIII 

15.0 

833 

12.0 

667 

IO.O 

556 

8.6 

476 

7-5 

417 

20 


SURVEYING. 


Table  XL*  is  such  a  table,  computed  for  total  centre  heights 
from  i  to  50  feet,  and  for  total  widths  from  I  to  100  feet. 
In  railroad  work  neither  of  these  quantities  can  be  as  small  as 
one  foot,  but  the  table  is  designed  for  use  in  all  cases  where, 
the  parallel  end  areas  may  be  subdivided  into  an  equal  number 
of  triangles  or  quadrilaterals. 

EXAMPLE  I.  Three- level  Ground  having  two  Warped  Surfaces. — Find  the 
volume  of  two  prismoids  of  which  the  following  are  the  field-notes,  the  width 
of  bed  being  20  feet,  and  the  slopes  i£  to  i. 


Station  n. 


Station  12. 


Station  12  -f-  5^. 


28.9f 


43-0 


+  12.6 

+  18.6 

4-22.0 

27.1 

0 

40.3 

4-11.4 

4-14.8 

4~  20.  2 

24.3 

0 

34-9 

4-9.5 


IO-3 


4- 


From  the  table,  p.  410,  giving  values  of  C0  and  K,  we  find  for 
and  r  =  i|,  C0  =  6. 7,  and  K  =  247. 

The  computation  may  be  tabulated  as  follows: 


=  20, 


Sta. 

Width, 
D=d+d'. 

Height, 
C  =  c  +  c0. 

Partial  Volume. 

Volume  of 
Prismoid. 

II 

71.9 

25-3 

562 

M 

69.6 

23-4 

503   X  4  =  2012 

12 

67.4 

21.5 

447 

3021  —  247 

2774 

M 

63.3 

19.2 

374  X  4  =  I496 

12  4-  56 

59-2 

17.0 

3ii 

.56(2254  -  247) 

1124 

*  Modeled  somewhat  after  Crandall's  Tables,  but  adapted  to  give  volumes 
by  the  Prismoidal  Formula  at  once  instead  of  by  the  method  of  mean  end  areas 
first  and  correcting  by  the  aid  of  another  table  to  give  prismoidal  volumes,  as 
Prof.  Crandall  has  done. 

f  The  numerators  are  the  distances  out,  and  the  denominators  are  the  heights 
above  grade,  4-  denoting  cut  and  —  fill. 


THE  MEASUREMENT  OF    VOLUMES. 


21 


Entering  the  table  (No.  XI.)  for  a  width  of  71  and  a  height  of  25,  we  find 
548,  to  which  add  7  for  the  3  tenths  of  height,  and  7  more  for  the  9  tenths  in 
width,  both  mentally,  thus  giving  562  cu.  yds.  for  this  partial  volume.  Simi- 
larly for  the  width  67. 4,  and  height  21.5,  obtaining  447  cu.  yds.  The  correspond- 
ing result  for  the  middle  area  is  503,  which  is  to  be  multiplied  by  4,  thus  giving 
2012  cu.  yds.  The  sum  of  these  is  3021  cu.  yds.,  from  which  is  to  be  subtracted 
the  constant  volume  K,  which  in  this  case  is  247  cu.  yds.,  leaving  2774  cu.  yds. 
as  the  volume  of  the  prismoid. 

The  next  prismoid  is  but  56  feet  long,  but  it  is  taken  out  just  the  same  as 
though  it  were  full,  and  then  56  hundredths  of  the  resulting  volume  taken. 
The  data  for  the  I2th  station  is  used  in  getting  this  result  without  writing  it 
again  on  the  page. 

EXAMPLE  2.  Five-level  Ground  having  four  Warped  Surfaces. — Find  the 
volume  of  a  prismoid  of  which  the  following  are  the  field-notes,  the  width  of 
bed  being  20  feet,  and  the  slopes  i?  to  I : 


II. 


28.9 


15-0 


+  12.6         +12.0        +18.6 


20.0 
+  21.0 


43-0 
+  22. 0 


12. 


27.1 


12.5 


+  II.4  +12.0  +  14- 


18.5  40-3 

+  19.6  +20.2 


This  is  the  same  problem  as  the  preceding,  with  intermediate  heights 
added. 

To  compute  this  from  the  table,  it  is  separated  into  three  prismoids,  as  shown 
in  Fig.  113. 


Let  ABDGCFE  be  the  cross-section.  This  may  be  separated  into  the  triangle 
ABC,  and  the  two  quadrilaterals  BCGD  and  ACFE.  The  area  of  the  triangle  is 
%cw.  That  of  the  right  quadrilateral  is,  from  Art.  179,  p.  202, 


22 


SURVEYING. 


~  o) 


Similarly  the  area  of  the  left  quadrilateral  is       £    (*—  /£')(  tfk  —  —  )  +  £'</* 


The  total  area  of  the  section  then  is 


-^  J    |. 


(i) 


If   the   interior  side  elevations    be  taken    over   the   edges  of   the   base,   then 
d  k  --   and  dk  --   both  become  zero,  and  the  first  and  last  terms  disappear. 

Or  if  the  centre  and  extreme  side  heights  are  the  same,  these  terms  go  out. 
Experience  shows  that  these  terms  can  usually  be  neglected  without  material 
error.  If  they  are  retained,  each  partial  volume  will  be  composed  of  five  terms, 
while  if  they  are  neglected  there  will  be  but  three.  The  signs  of  these  terms  also 
must  be  carefully  attended  to.  When  the  interior  side  readings  are  taken  over  the 
edges  of  the  base,  therefore,  this  equation  becomes 


A-\(k'd\  + 


(2) 


The  tables  are  well  adapted  to  compute  the  prismoidal  volume  for  five-level 
sections  by  either  of  these  formulae.  Thus,  if  the  adjacent  section  also  has  five 
points  determined  in  its  surface,  its  area  may  be  represented  by  an  equation  similar 
to  one  of  these,  and  from  these  end-area  data  mean  values  may  be  found  for  the 
corresponding  middle-area  points,  and  the  volumes  taken  out  as  before.  In  this 
case  the  prism  included  between  the  road-bed  and  side- slopes,  whose  volume  is  K, 
is  not  included,  and  hence  its  volume  is  not  to  be  deducted  from  the  result.  The 
computation  by  table  XI.  of  equation  (i)  would  be  as  follows  ; 


Sta. 

k'. 

d'h- 

k'. 

d'k. 

e. 

4- 

I 

dh. 

A. 

Partial  Volumes. 

Total 
Volume. 

ii 

12.6 

28.9 

12.0 

15.0 

18.6 

20.0 

21.043.0 

22.  0 

4-9  +  108  +  114  +  279—10=  500 

M 

12.0 

28.0 

12.  0 

13.8 

16.7 

rq.2 

20.3141.6 

21.  I 

4(  +  6  +  104  +  102  +  260  -  12)=  1840 

12 

11.427.* 

12.0 

«.5 

14-8 

18.5 

19.  6  40.3 

20,2 

+  3  +  100+  90  +  242—13  =  422 

2762 

•     1 

THE  MEASUREMENT  OF    VOLUMES. 


The  use  of  the  table  is  the  same  as  before.  First  take  out  from  the  table  the 
volume  corresponding  to  (c  —  h')(d'k  —  —  J,  which  when  evaluated  for  section  n 

is  (18.6  —  12.6)  (15.0—  10)  =  6.0  x  5.0.  This  is  positive,  and  the  volume  corre- 
sponding to  a  depth  of  6.0  feet  and  a  width  of  5.0  feet  is  9  cubic  yards.  Proceed 
to  evaluate  the  remaining  terms  of  eq.  (i)  in  a  similar  manner,  the  last  term 
coming  out  negative.  The  dimensions  of  the  mid  section  are  the  means  of  the 
corresponding  end  dimensions,  as  before.  If  one  end-area  is  a  three- level  section 
and  the  next  a  five-level  section,  the  included  prismoid  is  computed  as  a  five-level 
prismoid,  the  vanishing  points  in  the  three-level  section  corresponding  to  the 
interior  side  elevations  on  the  five-level  section  being  indicated  in  the  field.  Par- 
tial stations,  or  prismoids,  are  first  computed  as  though  they  were  100  feet  long 
(for  which  the  table  is  constructed),  and  then  multiplied  by  their  length  and  divided 
by  100  as  before. 

If  equation  (2)  may  be  used,  the  work  is  shortened  very  much.  The  columns 
in  h! ',  d'ic ,  dk  ,  and  h>  may  be  omitted,  and  there  will  also  be  but  three  terms  in 
each  partial  product.  Thus,  if  sections  n  and  12  had  been  taken  with  the  interior 
elevations,  each  10  feet  from  the  centre  line,  we  might  have  had  something  as 
follows  : 


n. 


28.9 

+  12.6 


IO.O 

+  15.4 


+  18.6 


IO.O 

19.* 


43.0 

+  22.0 


12. 


27-1 


IO.O 


+-11.4         +12.5         +14. 


10.0  4°-3 

+  17.4  +20.2 


The  computation  then,  by  eq.  (2),  would  have  been  : 


Sta. 

d'h. 

k>. 

C. 

k. 

dk. 

Partial  Volumes. 

Total 
Volume. 

II 

28.9 

15-4 

18.6 

19.8 

43-0 

137  +  114  +  263    =     5H 

II 

28.  c 

I4.O 

16.7 

18.6 

41.6 

4  (121  +  102   +  239)  =  1848 

12 

27.3 

12-5 

14.8 

17.4 

40.3 

IO4  +     90  +  215     =     409 

2771 

By  this  method  the  computation  of  a  five-level  section  is  little  more  trouble 


24 


SURVEYING. 


than  that  of  a  three-level  section,  and  yet  the  intermediate  points  taken  at  a  dis- 

w 
tance   of  —  from  the   centre,   are  apt  to  increase  the  accuracy  considerably  on 

ordinary  rolling  ground. 

321.  Three-level  Sections,  the  Surface  divided  into 
four  Planes  by  Diagonals. — If  the  surface  included  between 
two  three-level  sections  be  assumed  to  be  made  up  of  four 
planes  formed  by  joining  the  centre  height  at  one  end  with  a 
side ,  height  at  the  other  end  sec- 
tion on  each  side  the  centre  line 
(Fig.  114),  these  lines  being  called 
diagonals,  an  exact  computation  of 
the  volume  is  readily  made  without 
computing  the  mid-area.  Two  diag- 
onals are  possible  on  each  side  the 
centre  line  but  the  one  is  drawn 
which  is  observed  to  most  nearly 
fit  the  surface.  They  are  noted  in 
the  field  when  the  cross-sections  are 
taken. 


FIG. 


The  total  volume  of  such  a  prismoid  in  cubic  *  yards  is 


_-_ 


DC+  D'C 


where  clt 
tion  and 


(i) 


and  h{  are  the  centre  and  side  heights  at  one  sec- 
and d^  the  distances  out,  c^  h%,  h^  d^  and  d^  be- 


*  For  a  demonstration  of  this  formula  see  Henck's  Field-Book. 


THE   MEASUREMENT  OF    VOLUMES. 


ing  the  corresponding  values  for  the  other  end  section.  C  and 
C  are  the  centre  heights,  H  and  H'  the  side  heights,  and  D 
and  D'  the  distances  out  on  the  right  and  left  diagonals. 
Although  this  formula  seems  long,  the  computations  by  it  are 
very  simple.  Thus  let  the  volume  be  found  from  the  following 
field-notes  for  a  base  of  20  feet  and  side  slopes  i£  to  i. 


22 


+  16 


+  4 


The  upper  figures  indicate  the  distances  out  and  those 
below  the  lines  the  heights,  the  plus  sign  being  used  for  cuts* 
The  computation  in  tabular  form  is  as  follows : 


Sta. 

d. 

h. 

c. 

A'. 

d'. 

<*+</'. 

v+** 

DC. 

zxc*. 

I 

#22 

8 

8 

25 

47-5 

69.5 

556 

.... 

.... 

2 

34 

16 

4 

4 

16 

50.0 

200 

88 

128 

hi  +  k*  -  24                  88 

tf-\-tf'=I2                        128 

IV  ^-, 

—  Sh's  =  65  X  ^o            =  650 

6  )  162200 

27 )  27033 

TOOI  cu.  yards. 

The  great  advantage  of  the  method  consists  in  the  datai 
all  being  at  hand  in  the  field-notes. 

Hudson's  Tables  *  give  volumes  for  this  kind  of  prismoid. 

*  Tables  for  Computing  the  Cubic  Contents  of  Excavations  and  Emoank- 
ments.     By  John  R.  Hudson,  C.E,     John  Wiley  &  Sons,  New  York,  1884. 


26  SURVEYING. 


They  furnish  a  very  ready  method  of  computing  volumes  wheu 
this  system  is  used. 

322.  Comparison  of  Methods  by  Diagonals  and  by 
Warped  Surfaces.— Although  the  surveyor  has  a  choice  oi 
two  sets  of  diagonals  when  this  method  is  used,  the  real  surface 
would  usually  correspond  much  nearer  the  mean  of  the  two  pairs 
of  plane  surfaces  than  to  either  one  of  them.  That  is,  the 
natural  surface  is  curved  and  not  angular,  and  therefore  it  is 
probable  that  two  warped  surfaces  joining  two  three-level  sec- 
tions would  generally  fit  the  ground  better  than  four  planes, 
notwithstanding  the  choice  that  is  allowed  in  the  fitting  of  the 
planes.  More  especially  must  this  be  granted  when  the  truth 
of  the  following  proposition  is  established. 

PROPOSITION :  The  volume  included  between  two  three-level 
sections  having  their  corresponding  surface  lines  joined  by 
warped  surfaces,  is  exactly  a  mean  between  the  two  volumes 
formed  between  the  same  end  sections  by  the  two  sets  of  planes  re- 
sulting from  the  two  sets  of  diagonals  which  may  be  drawn. 

If  the  two  sets  of  diagonals  be  drawn  on  each  side  the 
centre  line  and  a  cross-section  be  taken  parallel  to  the  end 
areas,  the  traces  of  the  four  surface  planes  on  each  side  the 
centre  line  on  the  cutting  plane  will  form  a  parallelogram, 
the  diagonal  of  which  is  the  trace  of  the  warped  surface  on 
this  cutting  plane.  Since  this  cutting  plane  is  any  plane  par- 
allel to  the  end  areas,  and  since  the  warped  surface  line  bisects 
the  figure  formed  by  the  two  sets  of  planes  formed  by  the 
diagonals,  it.  follows  that  the  warped  surface  bisects  the  volume 
formed  by  the  two  sets  of  planes.  The  proposition  will  there- 
fore be  established  if  it  be  shown  that  the  trace  of  the  warped 
surface  is  the  diagonal  of  the  parallelogram  formed  by  the 
traces  of  the  four  planes  formed  by  the  two  sets  of  diagonals. 
Fig.  115  shows  an  extreme  case  where  the  centre  height  is 
higher  than  the  side  height  at  one  end  and  lower  at  the  other. 
Only  the  left  half  of  the  prismoid  is  shown  in  the  figure.  The 


THE  MEASUREMENT  OF    VOLUMES. 


cutting  plane  cuts  the  centre  and  side  lines  and  the  two  diago- 
nals in  efgh  on  the  plane,  and  in  e'f'g'h'  on  the  vertical 
projection.  For  the  diagonal  c^  the  surface  lines  cut  out  are 
//'  and  f'ti.  For  the  diagonal  c^  they  are  e'g'  and  g'k'. 
For  the  warped  surface  the  line  cut  out  is  e'h ',  this  being  an 


FIG.  115. 

element  of  that  surface.     It  remains  to  show  that  e'f'tig'  is  a 
parallelogram. 

Since  the  cutting  plane  is  parallel  to  the  end  planes  all  the 
lines  cut  are  divided  proportionally.  That  is,  if  the  cutting 
plane  is  one  nih  of  /  from  c»  then  it  cuts  off  one  wth  of  all  the 
lines  cut,  measured  from  that  end  plane.  But  if  the  lines 
are  divided  proportionally,  the  projections  of  those  lines  are 
divided  proportionally,  and  hence  the  points  e'>ff>h',2'  divide 


28  SURVEYING. 


the  sides  of  the  quadrilateral  */„',  */,  r/X'  proportionally.  But 
it  is  a  proposition  in  geometry  that  if  the  four  sides  of  a  quad- 
rilateral, or  two  opposite  sides  and  the  diagonals,  be  divided 
proportionally  and  the  corresponding  points  of  subdivision 
joined,  the  resulting  figure  is  a  parallelogram.  Therefore  ef'tf 
g'  is  a  parallelogram,  and  e'h'  is  one  of  its  diagonals  and  hence 
bisects  it.  Whence  the  surface  generated  by  this  line  moving 
along  £•/•„  and  d^  parallel  to  the  end  areas  bisects  the  volume 
formed  by  the  four  planes  resulting  from  the  use  of  both  di- 
agonals on  one  side  the  centre  line.  Q.  E.  D. 

It  is  probable,  therefore,  that  the  warped  surface  would 
usually  fit  the  ground  better  than  either  of  the  sets  of  planes 
formed  by  the  diagonals.  Furthermore,  the  errors  caused  by 
the  use  of  the  warped  surface  (Table  XI.)  are  compensating 
errors,  thus  preventing  any  marked  accumulation  of  errors  in 
a  series  of  prismoids.*  There  are  extreme  cases,  however, 
such  as  that  given  in  the  example,  Fig.  1 14,  which  are  best 
computed  by  the  method  by  diagonals. 

323.  Preliminary  Estimate  from  the  Profile. — If  the 
cross-sections  be  assumed  level  transversely  then  for  given 
width  of  bed  and  side  slopes,  a  table  of  end  areas  may  be  pre- 
pared in  terms  of  the  centre  heights.  From  such  a  table  the 


*  The  two  methods  here  discussed  are  the  only  ones  that  have  any  claims  to 
accuracy.  The  method  by  "  mean  end  areas,"  wherein  the  volume  is  assumed 
to  be  the  mean  of  the  end  areas  into  the  length,  always  gives  too  great  a  volume 
(except  when  a  greater  centre  height  is  found  in  connection  with  a  less  total 
width,  which  seldom  occurs),  the  excess  being  one  sixth  of  the  volume  of  the 
pyramids  involved  in  the  elementary  forms  of  the  prismoid.  This  is  a  large  error 
even  in  level  sections,  and  very  much  greater  on  sloping  ground,  and  yet 
it  is  the  basis  of  most  of  the  tables  used  in  computing  earthwork,  and  in  some 
States  it  is  legalized  by  statute.  Thus  in  the  example  computed  by  Henck's 
method  on  p.  414  the  volume  by  mean  end  areas  is  1193  cu.  yards;  by  the 
prismoidal  formula  it  is  1168  cu.  yards,  while  by  the  method  by  diagonals  it  was 
only  1001  cu.  yards.  This  was  an  extreme  case,  however,  and  was  selected  to 
show  the  adaptation  of  the  method  by  diagonals  to  such  a  form. 


UNIVERSITY 


THE  MEASUREMENT  OF   VOLUMES.  29 

end  areas  may  be  rapidly  taken  out  and  plotted  as  ordinates 
from  the  grade  line.  The  ends  of  these  ordinates  may  then 
be  joined  by  a  free-hand  curve,  and  the  area  of  this  curve 
found  by  the  planimeter.  The  ordinates  may  be  plotted  to 
such  a  scale  that  each  unit  of  the  area,  as  one  square  inch, 
shall  represent  a  convenient  number  of  cubic  yards,  as  1000. 
The  record  of  the  planimeter  then  in  square  inches  and  thou- 
sandths gives  at  once  the  cubic  yards  on  the  entire  length  of 
line  worked  over  by  simply  omitting  the  decimal  point.  Evi- 
dently the  scale  to  which  the  ordinates  are  to  be  drawn  to  give 
such  a  result  is  not  only  a  function  of  the  width  of  bed  and 
side  slopes,  but  also  of  the  longitudinal  scale  to  which  the  pro- 
file line  is  plotted.  The  area  of  a  level  section  is 

A  =wc  +  rc\ (I) 

where  w,  c,  and  r  are  the  width  of  base, .  centre  height,  and 
slope-ratio  respectively. 

Now  if  h  =  the  horizontal  scale  of  the  profile,  that  is  the 
number  of  feet  to  the  inch,  and  if  one  square  inch  of  area  is  to 
represent  1000  cu.  yards,  the  length  of  the  ordinate  must  be 

-         hA          -  k(M  +  r<?) 
1000  X  27  ~       27,000 

If  values  be  given  to  h,  w,  and  r,  which  are  constants  for 
any  given  case,  then  the  value  of  y  becomes  a  function  of  c 
only,  and  a  table  can  be  easily  prepared  for  the  case  in  hand. 
Since  y  is  a  function  of  the  second  power  of  cy  the  second  dif- 
ference will  be  a  constant,  and  the  table  can  be  prepared  by 
means  of  first  and  second  differences.  Thus  if  c  takes  a  small 
increment,  as  i  foot,  then  the  first  difference  is 


SUR  VE  YING. 


But  this  first  difference  is  also  a  function  of  c,  and  hence  when 
c  takes  an  increment  this  first  difference  changes  by  an  amount 
equal  to 


h 

27000 


(4) 


which  is  constant.  An  initial  first  difference  being  given  for  a 
certain  value  of  c,  a  column  of  first  differences  can  be  obtained 
by  simply  adding  the  A'  'y  continuously  to  the  preceding  sum. 
With  this  column  of  first  differences  the  corresponding  column 
of  values  of  y  may  be  found  by  adding  the  first  differences  con- 
tinuously to  the  initial  value  of  y  for  that  column.* 


TABULAR  VALUES  OF  y  IN  EQUATION  (2)  FOR  w=2o,  r= 

h  =  400. 


,  AND 


c 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

o 

o.oo 

0.03 

0.06 

0.09 

O.  12 

0.15 

o.  19 

O.22 

0.25 

0.28 

I 

•32 

•35 

•39 

.4$ 

.46 

•49 

•53 

•57 

.61 

.64 

2 

.68 

.72 

.76 

.80 

.84 

.88 

.92 

.96 

1.  00 

1.05 

3 

1.09 

*-»3 

1.17 

1  .22 

1.26 

*-3* 

i-35 

1.40 

I-45 

1.49 

4 

i-54 

i-59 

1.63 

1.69 

i-73 

1.78 

1.83 

1.88 

i-93 

1.99 

5 

2.04 

2.09 

2.14 

2.  19 

2.24 

2.30 

2.36 

2.41 

2-47 

3-52 

6 

2.58 

2.63 

2.69 

2-75 

2.80 

2.87 

2.92 

2.98 

3-°4 

3.10 

7 

3-i6 

3-22 

3-28 

3-35 

3-41 

3  47 

3-54 

3-60 

3-66 

3-73 

8 

3-79 

3-86 

3-92 

3-99 

4-05 

4-i3 

4.19 

4.26 

4-33 

4.40 

9 

4-47 

4-54 

4.60 

4.68 

4-75 

4.82 

4.89 

4-97 

5-°4 

5-n 

10 

5.18 

5-26 

5-33 

5-40 

5  48 

5.56 

5-64 

5-72 

5-79 

5.87 

it 

5-95 

6.03 

6.10 

6.18 

6.26 

6-35 

6.43 

6.51 

6-59 

6.67 

12 

6.76 

6.84 

6.92 

7.00 

7.09 

7.18 

7.26 

7-35 

7-43 

7-52 

13 

7.61 

7.70 

7.78 

7.86 

796 

8.05 

8.14 

8.23 

8.32 

8.41 

M 

8.50 

8.60 

8.68 

8-77 

8.87 

8.97 

9.06 

9.  16 

9-25 

9-35 

15 

9-44 

9-54 

9-63 

9-73 

9-83 

9-94 

10.03 

10.13 

10.23 

10.33 

16 

10.43 

io-53 

10.62 

10.73 

10.83 

10.94 

ii  .04 

11.15 

".25 

"•35 

J7 

11.46 

11.56 

11.66 

11.77 

11.88 

12.00 

12.10 

12.21 

12.31 

12.42 

18 

12.53 

12.64 

12.75 

12.86 

12.97 

13.09 

13.20 

J3-32 

13.42 

13-54 

'9 

13-65 

»3-77 

13-87 

13-99 

14.  10 

14  23 

14-34 

14.47 

14-58 

14.70 

20 

I4.&I 

14-93 

15.04 

15.16 

TS.aq 

X5,^ 

15-53 

15-66 

1=5-78 

15.90 

*  For  a  further  exposition  of  this  subject,  see  Appendix  C. 


THE   MEASUREMENT  OF    VOLUMES. 


The  preceding  table  was  constructed  in  this  manner,  for 
w  =  20  feet,  r  =  i£ ;  and  h  —  400  feet  to  the  inch. 

324.  Borrow-pits  are  excavations  from  which  earth  has 
been  "  borrowed  "  to  make  an  embankment.     It  is  generally 
preferable  to  measure  the  earth  in  cut  rather  than  in  fill,  hence 
when  the  earth  is  taken  from  borrow-pits  and  its  volume  is  to 
be  computed  in  cut,  the  pits  must  be  carefully  staked  out  and 
elevations  taken  both  before  and  after  excavating.     The  meth- 
ods given  in  art.  311  are  well  suited  to  this  purpose,  or  they 
may  be  computed  as  prismoids  by  the  aid  of  Table  XL,  if  pre- 
ferred.    To  use  the  table  it  is  only  necessary  to  enter  it  with 
such  heights  and  widths  as  give  twice  the  elementary  areas 
(triangles  or  quadrilaterals)  into  which  the  end  sections  are 
divided,  and  then  multiply  the  final  result  by  the  length  and 
divide  by  100.     The  table  is  entered  for  both  end-area  dimen- 
sions and  also  the  mid-area  dimensions,  four  times  this  latter 
result  being  taken  the  same  as  before. 

325.  Shrinkage   of  Earthwork. — Excavated    earth    first 
increases  in  volume,  when  removed  from  a  cut  and  dumped  on 
a  fill,  but  it  gradually  settles,  or  shrinks,  until  it  finally  comes 
to  occupy  a  less  volume  than  it  formerly  did  in  the  cut.     Both 
the  amounts,  initial  increase,  and  final  shrinkage  depend  on  the 
nature  of  the  soil,  its  condition  when  removed,  and  the  man- 
ner of  depositing  it  in  place.     There  can  therefore  be  no  gen- 
eral rules  given  which  will  always  apply.     For  ordinary  clay 
and  sandy  loam,  dumped  loosely,  the  first  increase  is  about  one 
twelfth,  and  then  the  settlement  about  one  sixth  of  this  increased 
volume,  leaving  a  final  volume  of  about  nine  tenths  of  the  original 
volume  in  cut.* 

Thus  for  100  cubic  yards  of  settled  embankment  in  cubic 
yards  in  cut  would  be  required.     But  a  contractor  should  have 

*  See  paper  by  P.  J.  Flynn  in  Trans.  Tech.  Soc.  of  the  Pacific  Coast,  voi 
ii.  p.  179,  where  all  the  available  experimental  data  are  given. 


32  SURVEYING. 


his  stakes  or  poles  set  one  fifth  higher  than  the  corresponding 
fill,  so  that  when  filled  to  the  tops  of  these,  a  settlement  of 
one  sixth  will  bring  the  surface  to  the  required  grade. 

These  changes  of  volume  are  less  for  sand  and  more  for 
stiff,  wet  clay. 

For  rock  the  permanent  increase  in  volume  is  from  60  to 
80  per  cent,  the  greater  increase  corresponding  to  a  smaller 
average  size  of  fragment. 

326.  Excavations  under  Water. — It  is  often  necessary  to 
determine  the  volume  of  earth,  sand,  mud,  or  rock  removed 
from  the  beds  of  rivers,  harbors,  canals,  etc.  If  this  be  done 
by  soundings  alone,  it  is  likely  to  work  injustice  to  the  con- 
tractor, as  he  would  receive  no  pay  for  depths  excavated  below 
the  required  limit ;  and  besides,  foreign  material  is  apt  to  flow 
in  and  partially  replace  what  is  removed,  so  that  the  material 
actually  excavated  is  not  adequately  shown  by  soundings 
within  the  required  limits.  It  is  common,  therefore,  to  pay 
for  the  material  actually  removed,  an  inspector  being  usually 
furnished  by  the  employer  to  see  that  no  useless  work  is  done 
beyond  the  proper  bounds.  The  material  is  then  measured  in 
the  dumping  scows  or  barges.  The  unit  of  measure  is  the 
cubic  yard,  the  same  as  in  earthwork.  There  are  two  general 
methods  of  gauging  scows,  or  boats.  One  is  to  actually  meas- 
ure the  inside  dimensions  of  each  load,  which  is  often  done  in 
the  case  of  rock,  and  the  other  is  to  measure  the  displacement 
of  the  boat,  which  is  the  more  common  method  with  dredged 
material.  When  the  barge  is  gauged  by  measuring  its  dis- 
placement, the  water  in  the  hold  must  always  be  pumped  down 
to  a  given  level,  or  else  it  must  be  gauged  both  before  and  after 
loading  and  the  depth  of  water  in  the  hold  observed  at  each 
gauging.  A  displacement  diagram  (or  table)  is  prepared  for 
each  barge,  from  its  actual  external  dimensions,  in  terms  of  its 
mean  draught.  There  should  always  be  four  gaugings  taken 
to  determine  the  draught,  at  four  symmetrically  located  points 


THE  MEASUREMENT  OF    VOLUMES.  33 

on  the  sides,  these  being  one  fourth  the  length  of  the  barge 
from  the  ends.  Fixed  gauge-scales,  reading  to  feet  and  tenths 
may  be  painted  on  the  side  of  the  barge,  or  if  it  is  flat-bot- 
tomed, a  gauging-rod,  with  a  hook  on  its  lower  end  at  the  zero 
of  the  scale,  may  be  used  and  readings  taken  at  these  four 
points.  Any  distortion  of  the  barge  under  its  load,  or  any 
unsymmetrical  loading,  will  then  be  allowed  for,  the  mean  of 
the  four  gauge-readings  being  the  true  mean  draught  of  the 
boat. 

To  prepare  a  displacement  diagram,  the  areas  of  the  sur- 
faces of  displacement  must  be  found  for  a  series  of  depths  uni- 
formly spaced.  This  series  may  begin  with  the  depth  for  no 
load,  the  hold  being  dry.  They  should  then  be  found  for  each 
five  tenths  of  a  foot  up  to  the  maximum  draught.  If  the  boat 
has  plane  vertical  sides  and  sloped  ends  these  areas  are  rec- 
tangles, and  are  readily  computed.  If  the  boat  is  modelled  to 
curved  lines,  the  water-lines  can  be  obtained  from  the  original 
drawings  of  the  boat,  or  else  they  must  be  obtained  by  actual 
measurement.  In  either  case  they  can  be  plotted  on  paper, 
and  their  areas  determined  by  a  planimeter.  These  areas  are 
analogous  to  the  cross-sections  in  the  case  of  railroad  earth- 
work, and  the  prismoidal  formula  may  be  applied  for  comput- 
ing the  displacement.  Thus, 

Let  A0,  Aiy  A^  A9J  etc.,  be  the  areas  of  the  displaced  water 
surfaces,  taken  at  uniform  vertical  distances  h  apart.  Then 
for  an  even  number  of  intervals  we  have  in  cubic  yards 


.   .    (i) 


If  the  total  range  in  draught  be  divided  into  six  equal  por- 
tions, each  equal  to  h,  then  Weddel's  Rule  *  would  give  a 


*  For  the  derivation  of  this  rule  see  Appendix  C. 
o 


34 


SUR  VE  YING. 


nearer  approximation.     With  the  same  notation  as  the  above 
we  would  then  have,  in  cubic  yards, 


(2) 


These  rules  are  also  applicable  to  the  gauging  of  reservoirs, 
mill-ponds,  or  of  any  irregular  volume  or  cavity. 

After  the  displaced  volume  of  water  is  found,  the  corre- 
sponding volume  of  earth  or  rock  is  found  by  applying  a  proper 
constant  coefficient.  This  coefficient  is  always  less  than  unity, 
and  is  the  reciprocal  of  the  specific  gravity  of  the  material. 
This  must  be  found  by  experiment.  In  the  case  of  soft  mud 
it  is  nearly  unity,  while  with  sand  and  rock  it  is  much  more. 
When  rock  is  purchased  by  the  cubic  yard,  solid  rock  is  not 
implied,  but  the  given  quality  of  cut  or  roughly-quarried  rock, 
piled  as  closely  as  possible.  When  rock  is  excavated,  solid 
rock  is  meant.  A  measured  volume  of  any  material  put  into  a 
gauged  scow  will  give  the  proper  coefficient  for  that  material. 
Thus  if  the  measured  volume  V  give  a  displacement  of  F, 

V 

then  -pr  =  C  is  the  coefficient  to  apply  to  the  displacement  to 

give  the  volume  of  that  material. 


TABLES. 


TABLE  I. 
TRIGONOMETRIC  FORMULAE. 


TRIGONOMETRIC  FUNCTIONS. 

Let  A  (Fig.  107)  =  angle  BAG  =  arc  BF,  and  let  the  radius  AF  —  AB 
AH=l. 
We  then  have 

H 


cos  4  =  AG 

tan  A  =  D F 

cot  A  =  HG 

sec  A  =  AD 

cosec  A  -AG 

versin  ^4  =  CF  =  BE 

covers  A  =  BK  =  HL 

exsec  A  =  BD 

coexsec  A  =  EG 

chord  A  =  BF 

chord  2  A  =  BI  =  2#<7 


In  the  right-angled  triangle  ABC  (Fig.  107) 
Let  ^4Z?  =  c,  AC  -  6,  and  BC  =  a. 
We  then  have : 


Fio.  107. 


1.  sin  A 

2.  cos  A 

3.  tan  A 

4.  cot  ^. 

5.  sec  A 


=      —      —  coaB 
c 


ss     —     'SB  sin  B 


-r~      =  cot  j5 
» 


—      =  tan  j? 
a 


=     -—     =  cosec  j5 


6.  cosec  A     —     —     =  sec  5 

a 

7.  vers  ^4       =  °  ~     —  covers  B 

8     exsec  A    =  —  —  =  coexsec  B 
o 


9.    covers^   = 


=  versin  B 


11.  a  =  c  sin  yl  =  6  tan  A 

12.  6  =  c  cos^4.  =  a  cot  .4 

13  c  -       a       =      & 

sin  A       cos  .4 

14.  a  =  c  cos  B  =  b  cot  B 

15.  6  =  c  sin  P  =  a  tan  B 

16.  c  .    ...o_  =  ..»_ 

17-  a  =  V  (C  +  6)  (c  -  6) 

18.  &  =  V(C  +  a)  (o  -  a) 

19.  c  = 

20.  C7  = 


21.  area  =  ^- 


SURVEYING. 


TABLE  I.— Continued. 
TRIGONOMETRIC  FORMUUE. 


SOLUTION  OF  OBLIQUE  TRIANGLES. 


FIG.  108. 


GIVEN.  SOUGHT. 


22 

88 

84 
US 

26 

27 

28 
29 

30 
31 

33 
33 


A,  B,a 


A,a,b 


C,  a,  6 


a,  6,  c 


C,  6,  c   i  C  =  180° 


b  = 


J?,  <?,   C 


area 
area 


sin  ^  =        -     .6.  C  =  180° 


C  =    -r—  r  .  Sin  C. 

sin  ^fl 


vers  A  =  — 


K 


V«  («  -  a)  («  -  6)  (J  -  c) 

a2  sin  5  .  sin  C 
2  sin'  A 


TABLES.  39 

TABLE  I. — Continued. 
TRIGONOMETRIC  FORMULAE. 


GENERAL  FORMULA. 


3-1 


sin  A    —     -       — ;     =     VI  —  cosa  A    =     tan  A  cos  A 
cosecA 

sin  A    =    2  sin  y^  A  cos  ^  A    - 


sin  A    -     y  levers  2  A    = 


cos  A    =    sec^     =     y  1  —  sin*  J.     =    cot  ^4  sin  ^ 
1-vers^    =     2cosa^^4  —  1    =    1  — 


cos  .4    =    cosa  J^  A  —  sin'  ^  ^l     =    V 

_^         =    silL^ 
cot  A  cos  ^. 


tan  .4    =     J      1          -  1     -     Vi-cog2^ ^L»A_ 

y   cos2  .4  cos^  "     l -f-  cos  2  A 

1  —  cos  2  A  vers  2  A 


cos 


1+cosZA 


^ 
—  CO&2A  vera2A 

cot  A    , 


vers  A    =    1  —  cos  A    =    sin  ^4  tan  ^  ^4    =    2  sin"  }g  yl 

vers  A    =    exsec  ^4  cos  A 

exsec  ^4    =    sec  A  —  I    =    tan  ^4  tan  ^  ^4 

sin^^    =     •/1-cos^ 


_ 1    =    cos^A  —  sin*  A 


40  SURVEYING. 


TABLE  I. — Continued. 
TRIGONOMETRIC  FORMULAE. 


GENERAL  FORMULAE. 
tan  A 


sin  ^4  cosec  .4  —  cot  .4 


57. 


1  -f^l  —  J^  vers  A      2+  Vs  (1  +  cos 
58.  vers  2  A  =  2  sin2  -4 


61.  sin  (^4  ±  B)  =  sin  ^1  .  cos  5  db  sin  B  .  cos  ^4 

62.  cos  (^4  ±  B)  =  cos  ^4  .  cos  B  =F  sin  >4  .  sin  J? 

63.  sin^4  +  sinB  =  2sm^M  +  B)cos^(^~J5) 

64.  sin  ^4  —  sin  B  =  2  cos  J^  (^1  +  B)  sin  J£  (A  —  B) 

65.  cos  A  +  cos  B  =  2  cos  ^  (^4  +  B)  cos  H(A  —  B 

66.  cosB  —  eo8^  =  2sm^C4  +  P)sin^(.4  —  B) 


67.  sin»  ^4  —  sin*  B  =  cosa  B  —  cos8  ^4  =  sin  (A  +  J5)  sin  (A  —  B> 

68.  cos8  A  —  sin8  J?  =  cos  (A  +  J5)  cos  (A  —  B) 


cos  ^4  .  cos  5 


70.  tan  4  -  tan  B  =  -ri°'(/~'B)n 
cos  ^4  .  cos  B 


TABLES. 


TABLE    II. 
FOR  CONVERTING  METRES,  FEET,  AND  CHAINS. 


METRES  TO  FEET. 

FEET  TO  METRES  AND  CHAINS. 

CHAINS  TO  FEET. 

Metres. 

Feet. 

Feet. 

Metres. 

Chains. 

Chains. 

Feet. 

I 

3.28087 

! 

0.304797 

0.0151 

0.01 

0.66 

2 

6.56174 

2 

0.609595 

.0303 

.02 

1.32 

3 

9.84261 

3 

0.914392 

•0455 

.03 

1.98 

4 

13.12348 

4 

1.219189 

.0606 

.04 

2.64 

5 

16.40435 

5 

1.523986 

.0758 

•05 

3-30 

6 

19.68522 

6 

1.828784 

.0909 

.06 

3.96 

7 

22  .  96609 

7 

2.I3353I 

.1061 

.07 

4.62 

8 

26.24695 

8 

2.438378 

.1212 

.08 

5.28 

9 

29.52732 

9 

2-743I75 

.1364 

.09 

5-94 

10 

32.80869 

10 

3-047973 

.1515 

.  IO 

6.60 

20 

65.61739 

20 

6.095946 

.3030 

.20 

13.20 

30 

98.42609 

30 

9.143918 

•4545 

•30 

19.80 

40 

131.2348 

40 

12.19189 

.6o6l 

.40 

26.40 

50 

164.0435 

50 

15.23986 

.7576 

•50 

33-00 

60 

196.8522 

60 

18.28784 

.9091 

.60 

39-6o 

70 

229.6609 

70 

2I.3358I 

I.  0606 

.70 

46.20 

80 

262.4695 

80 

24.38378 

I.2I2I 

.80 

52.80 

go 

295.2782 

90 

27.43175 

1.3636 

.90 

59-40 

JOO 

328.0869 

IOO 

30.47973 

I.5I5I 

I 

66.00 

200 

656.1739 

100 

60.95946 

3.0303 

2 

132 

300 

984.2609 

300 

91.43918 

4-5455 

3 

198 

400 

1312.348 

400 

121.9189 

6.0606 

4 

264 

500 

1640.435 

500 

152.3986 

7.5756 

5 

330 

600 

1968.522 

600 

182.8784 

9.0909 

6 

396 

700 

2296.609 

700 

2I3.358I 

10.606 

7 

462 

800 

2624.695 

800 

243.8378 

12.  121 

8 

528 

900 

2952.782 

900 

274.3175 

13.636 

9 

594 

IOOO 

3280.869 

IOOO 

304.7973 

15.151 

IO 

660 

2000 

6561.739 

2000 

609  .  5946 

30-303 

20 

1320 

3000 

9842  .  609 

3000 

914.3918 

45-455 

30 

1980 

4OOO 

13123.48 

4OOO 

1219.189 

60.606 

40 

2640 

5000 

16404.35 

5000 

1523.986 

75-756 

50 

3300 

6OOO 

19685.22 

6000 

1828.784 

90,909 

60- 

396o 

7000 

22966.09 

7OOO 

2133.581 

106.06 

70 

4620 

8000 

26246.95 

8000 

2438.378 

121.  21 

80 

5280 

9000 

29527.82 

9000 

2743.175 

136.36. 

90 

5940 

SURVEYING. 


TABLE  III. 
LOGARITHMS  OF  NUMBERS.     §  173. 


Z   0 

a 

1 

a 

3 

4 

5 

6 

^ 

8 

9 

Proportional  Parts. 

1 

» 

3 

4 

5 

G 

7 

8 

9 

10   .0000 

.0043 

.0086 

.0128 

.0170 

.0212 

•  0253 

.0294 

•°334 

•0374 

4 

8 

12 

17 

21 

25 

29 

33 

37 

II  .0414 

•°453 

.0492 

.0531 

.0569 

.0607 

.0645 

.0682 

.0719 

•0755 

4 

8 

II 

19 

23 

26 

30 

34 

12   .0792 

.0828 

.0864 

.0899 

•°934 

.0969 

.  1004 

.1038 

.1072 

.  1106 

3 

7 

IO 

4 

21 

24 

28 

31 

I3   .1139 

•"73 

.1206 

.1239 

.1271 

•1303 

•  1335 

•  1367 

.1399 

.1430 

3 

6 

10 

16 

*9 

23 

26 

20 

14   .1461 

.1492 

•  1523 

•1553 

.1584 

.1614 

.1644 

•1673 

•I7°3 

•^S2 

3 

6 

9 

2 

15 

18 

21 

24 

27 

IS   .1761 

.1790 

.1818 

.1847 

•1875 

.1903 

•1931 

•1959 

.1987 

.2014 

3 

6 

8 

i 

M 

r7 

20 

22 

25 

i6(  .2041 

.2068 

•2095 

.2122 

.2148 

•2175 

.2201 

.2227 

.  2480 

•2253 

.2279 

3 

5 

8 

I 

13 

16 

18 

21 

2O 

24 

18  .2553 

•2577 

•^355 

.260! 

.2625 

^2648 

.2430 
.2672 

•2455 
•  2695 

.2718 

.25O4 

.2742 

.2529 

.2765 

5 

7 

9 

12 

16 

21 

19  .2788 

.2810 

•2833 

.2856 

.2878 

.2900 

•  2923 

•2945 

.2967 

.2989 

4 

7 

9 

11 

i3 

16 

18 

20 

20   .3010 

.3032 

•3°54 

•3°75 

•  3096 

.3118 

•3139 

3160 

•  3181 

•  3201 

4 

6 

8 

11 

13 

15 

17 

19 

21   .3212 

•3243 

•  3263 

•3284 

•33°4 

•3324 

•3345 

•3365 

•3385 

•34°4 

4 

6 

8 

10 

12 

H 

16 

si 

22   .3424 

•3444 

•3464 

•3483 

.3502 

•3522 

•3541 

•3560 

•3579 

•3598 

4 

6 

8 

10 

12 

14 

15 

17 

23   .3617 

.3636 

•3655 

•3674 

-3692 

•37" 

•3729 

•3747 

.3766 

•3784 

4 

6 

7 

9 

II 

13 

15 

17 

24   .3802 

.3820 

.3838 

•3856 

.3874 

•39°9 

•3927 

•3945 

.3962 

4 

5 

7 

9 

II 

12 

H 

16 

25  -3979 

•3997 

.4014 

.4031 

.4048 

•  4065 

.4082 

.4099 

.4116 

•4133 

3 

5 

7 

S 

10 

12 

I4 

15 

26  .4150 

.4166 

.4183 

.4200 

.4216 

.4232 

.4249 

.4^65 

.4281 

.4298 

3 

5 

7 

8 

10 

II 

13  15 

27  -43J4 

•433° 

•4346 

.4362 

.4378 

•4392 

.4409 

•4425 

.4440 

.4456 

3 

5 

6 

8 

9 

II 

13 

H 

28  .4472 

.4487 

.4502 

.4518 

•4504 

•4579 

•4594 

.4609 

3 

s 

6 

8 

9 

II 

12 

]4 

29  .4624 

•4639 

.4654 

.4669 

•  4683 

.4698 

•47I3 

.4728 

•4742 

•4757 

3 

4 

6 

7 

9 

10 

12 

13 

3°  -4771 

.4786 

.4800 

•4814 

.4829 

•4843 

•4857 

.4871 

.4886 

.4900 

3 

4 

6 

7 

9 

10 

II 

«3 

31  -49J4 

.4928 

.4942 

•4955 

.4969 

•4983 

•4997 

•  5011 

.5024 

.5038 

3 

4 

6 

7 

8 

10 

II 

12 

32  .5051 

•5065 

•5079 

.5092 

•5105 

•5"9 

•5132 

•5145 

•5159 

•S1?2 

3 

4 

5 

7 

8 

9 

II 

12 

33  -5185 

.5198 

.5211 

.5224 

•5237 

•  5250 

.5263 

.5276 

•5289 

•53°2 

3 

4 

5 

6 

8 

9 

10 

12 

34  -53*5 

•5328 

•534° 

•5353 

.5366 

•5378 

•5391 

•54°3 

.5416 

.5428 

3 

4 

5 

6 

8 

10 

II 

35  -5441 

•5453 

.5465 

•5478 

•5490 

•  5502 

•5514 

•5527 

•5539 

•555' 

4 

5 

6 

7 

9 

10 

II 

36,  -5563 

•5575 

•5587 

•5599 

.5623 

•5635 

•5647 

.5658 

.5670 

'  4 

5 

6 

7 

S 

TO 

II 

37  -5682 
38  .5798 

•5694 
.5809 

•5705 
•  5821 

•  5717 
•5832 

.5729 
•5843 

•5740 
.5855 

:US 

•5763 
•5877 

3S 

.5786 
•5899 

'  7 

5 

6 
6 

7 
7 

8 
8 

9 

9 

10 

IO 

39  -59" 

•5922 

•5933 

•5944 

•5955 

.5966 

•5977 

.5988 

•5999 

.6010 

3 

4 

5 

7 

8 

9 

10 

40  .6021 

.6031 

.6042 

.6053 

.6064 

.6075 

.6085 

.6096 

.6107 

.6117 

3 

4 

5 

6 

8 

9 

IO 

41  .6128 

•  6138 

.6149 

.6160 

.6170 

.6180 

.6191 

.6201 

.6212 

.6222 

3 

4 

5 

6 

7 

8 

9 

42  .6232 
43  -6335 

.6243 
•6345 

.6253 
•6355 

.6263 
•6365 

.6274 

•6375 

.6284 
•6385 

.6294 
•  6395 

.6304 
.6405 

•63*4 
•  6415 

•6325 

.6425 

1-3 

4 
4 

5 

6 
6 

7 
7 

8 
8 

9 

9 

44  -6435 

.6444 

•6454 

.6464 

.6474 

.6484 

•6493 

.6503 

•6513 

.6522 

3 

4 

5 

6 

7 

8 

9 

45  -6532 

.6542 

•6551 

•  6561 

•657' 

.6580 

.6^90 

•6599 

.6609 

.6618 

3 

4 

s 

6 

7 

8 

Q 

46  .6628 

.6637 

.6646 

.6656 

.6665 

.6675 

.6684 

•  6693 

.6702 

.6712 

1  3 

4 

5 

6 

f 

7 

8 

47  -6721 

.6730 

•6739 

.6749 

.6758 

.6767 

.6776 

.6785 

.6794 

.6803 

.  :  3 

4 

5 

5 

6 

7 

1 

48  .6812 

.6821 

.6830 

•  6839 

.6848 

.6857 

.6866 

.6875 

.6884 

•  6893 

3 

4 

4 

5 

6 

7 

& 

4g  .6902 

.6911 

.6920 

.6928 

•6937 

.6946 

•6955 

.6964 

.6972 

.6981 

3 

4 

4 

5 

6 

7 

fa 

50  -6990 

.6998 

.7007 

.7016 

.7024 

•7°33 

.7042 

.7050 

•  7°59 

.7067 

3 

3 

4 

5 

6 

7 

8 

51  -7076 

•  7084 

.7093 

.7101 

.7110 

.7118 

.7126 

•7135 

•7H3 

•  7152 

3 

3 

4 

5 

6 

7 

8 

52  .7160 

.7168 

.7177 

•7185 

•7193 

.7202 

.7210 

.7218 

.7226 

•  7235 

2 

3 

4 

5 

6 

7  7 

53  -7243 

.7251 

•7259 

.7267 

•7275 

•7284 

.7292 

.7300 

.7308 

•73l6 

2 

3 

4 

5 

6 

6  7 

54  -7324 

•7332 

•7340 

•7348 

-735^ 

•7364 

•7372 

.7380 

.7388 

•7396 

i 

2 

3 

4 

5 

6 

V 

TABLES. 


43 


TABLE    III.— Continued. 
LOGARITHMS  OF  NUMBERS. 


!  ° 

R 
fc 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  Parts. 

1  2 

3  4 

5 

6 

7 

8 

9 

55  -7404 

.7412 

.7419 

.7427 

•7435 

•7443 

•  7451 

•7459 

.7466 

•7474 

3 

4 

5 

5 

6 

7 

56  .7482 

.7490 

•7497 

•7505 

•75*3 

.7520 

•  7528 

•  7536 

•7543 

•7551 

• 

4 

5 

5 

6 

7 

57  -7559 

.7566 

•7574 

.7582 

•7589 

•7597 

.7604 

.7612 

.7619 

.7627 

i 

4 

5 

5 

6 

7 

58  -7634 

.7642 

.7649 

•  7657 

.7664 

.7672 

.7679 

.7686 

.7604 

7701 

2 

4 

4 

5 

6 

7 

59  -7709 

.7716 

•7723 

•773^ 

•7738 

•7745 

.7752 

.7760 

.7767 

•7774 

3 

4 

4 

5 

6 

7 

60  .7782 

.7789 

.7796 

.7803 

.7810 

.7818 

.7825 

.7832 

•7839 

.7846 

3 

4 

4 

5 

6 

6 

61  .7853 

.7860 

.7868 

•7875 

.7882 

.7889 

.7896 

•79°3 

.7910 

.7917 

jj 

4 

4 

5 

6 

6 

62  .7924 
63  -7993 

•7931 
.8000 

•7938 
.8007 

•7945 
.8014 

•7952 
.8021 

£3 

.7966 
.8055 

•7973 
.8041 

.7980 
.8048 

•7987 
•8055 

g 

4 
4 

5 

5 

6 

5 

6 
6 

64  .8062 

.8069 

,8075 

.8082 

.8089 

.8096 

.8102 

.8109 

.8116 

.8122 

ji 

4 

5 

5 

6 

65  .8129 

.8136 

.8142 

.8149 

.8156 

.8162 

.8169 

.8176 

.8182 

.8189 

3 

4 

5 

5 

6 

66  .8195 

.8202 

.8209 

-8215 

.8222 

.8228 

.8235 

.8241 

.8248 

.8254 

! 

4 

5 

5l  6 

67  .8261 

.8267 

.8274 

.8280 

.8287 

.8293 

.8299 

.8306 

.8312 

.8319 

; 

4 

5 

5 

6 

68  .8325 

•8331 

.8338  .8344 

•8351 

•8357 

•8363 

.8370 

.8376 

.8382! 

J 

4 

5 

6 

69  .8388 
70  .8451 

•8395 

Q  t  p.* 

.8401 
8463 

.8407 
8470 

.8414 
8476 

.8420 
8482 

.8426 
8488 

.8432 

8  \C\A 

•8439 

.  8^OO 

•8445 
8506 

2 

4 

5 

6 

5 

71  -8513 
72  -8573 

•°457 

.8519 
••8579 

•  °4uj 
.8525 
.8585 

.0470 

•853' 
.8591 

•°47U 

.8537 

•8597 

.040^ 

•8543 
.8603 

:1SS 

•O494 

:llll 

^8561 
.8621 

lie 

I; 

4 
4 

5 
5 
5 

5 
5 

73  -8633 

.8639 

•8645 

.8651 

.8657 

.866-, 

.8669 

.8675 

.8681 

.8686 

2 

4 

5 

5 

74  .8692 

.8698 

.8704 

.8710 

.8716 

.8722 

.8727 

•8733 

•8739 

•8745 

2 

4 

5 

5 

75  -8751 

.8756 

.8762 

.8768 

.8774 

.8779 

.8785 

.8791 

.8797 

.8802 

2 

3 

4 

5 

5 

76  .8808 

.8814 

.8820 

.8825 

.8831 

•8837 

.8842 

.8848 

•  8854 

.8859! 

2 

3 

4 

5 

5 

77  .8865 

.8871 

.8876 

.8882 

.8887 

.8893 

.8899 

.8904 

.8910 

.8915! 

2 

3 

4 

4 

5 

78  .8921 
79  .8976 

.8927 
.8982 

.8932 
.8987 

.8938 
.8993 

.8943 

.809? 

.8949 

.900*: 

•  8954 

QOOQ 

.8960 

QOI  c: 

.8965 

GO2O 

.8971 

002^ 

j 

2 

3 

4 

4 

5 

80  .9031 

.9036 

.9042 

.9047 

vyiyu 
•9053 

.9058 

.  y*-^y 

.9063 

•yui  j 

.9069 

•  y^MK) 

.9074 

•ymg 

.9079 

1 

2 

3 

4 

4 

5 
5 

81  .9085 

.9090 

.9096 

.9101 

.9106 

.9112 

.9117 

.9122 

.9128 

•9133 

! 

2 

3 

4 

4 

5 

82  .9138 

•9143 

.9149 

•9!54 

•9*59 

.9165 

.9170 

•9175 

.9180 

.9186 

2 

2 

3 

4 

4 

5 

83  .9191 

.9196 

.920! 

.9206 

.9212 

.9217 

.9222 

.9227 

•  9232 

.9238 

2   2 

3 

4 

4 

5 

84  .9243 

.9248 

•9253 

.9258 

.9263 

.9269 

.9274 

.9279 

.9284 

.9289 

2 

3 

4 

4 

5 

85  .9294 

.9299 

.9304 

•9309 

•9315 

.9320 

•9325 

•9330 

•9335 

•9340 

2 

3 

4 

4 

c 

86  -9345 

•9350 

•9355 

.9360 

•9365 

•937° 

•9375 

.9380 

•9385 

•9390 

2 

3 

4 

4 

5 

87  '9395 

.9400 

•9405 

.9410 

•9415 

.9420 

•9425 

•943° 

•9435 

.9440 

°! 

2 

3 

3 

4 

4 

88  .9445 

•9450 

•9455  -9460 

•9465 

.9469 

•9474 

•9479 

.9484 

.9489 

O 

2 

3 

3 

4 

4 

89  .9494 

•9499 

.9504 

•9509 

•9513 

.9518 

•9523 

•9528 

•9533 

•9538 

0 

j 

2 

3 

3 

4 

4 

90  .9542 

•9547 

•9552 

•9557 

.9562 

.9566 

•9571 

•9576 

.9581 

.9586 

O 

, 

2 

2 

3 

3 

4 

4 

91  .9590 
92  .9638 

•9595 
•9643 

.9600 
.9647 

.9605 
.9652 

.9609 
•9657 

.9614 
.9661 

.9619 
.9666 

.9624 
.9671 

.9628 
•9675 

•9633 
.9680' 

°i 
ol 

; 

2 

2 
2 

3 
3 

3 
3 

4 
4 

4 

93  -9685 

.9689 

.9694 

.9699 

•9703 

.9708 

•9713 

.9717 

.9722 

.9727 

0 

1   2 

2 

3 

3 

4 

4 

94  -9731 

•9736 

•9741 

•9745 

•975° 

•9754 

•9759 

•9763 

.9768 

•9773 

O; 

2 

2 

3 

3 

4 

4 

95  -9777 

.9782 

.9786 

.9791 

•9795 

.9800 

.9805 

.9809 

.9814 

.9818 

0 

2 

2 

3 

3 

4 

4 

96  .9823 

.9827 

.9832 

.9836 

.9841 

•9845 

.9850 

•9854 

.9859 

.9863 

o1 

2 

2 

3 

4 

4 

97  .9868 

.9872 

.9877 

.9881 

.9886 

.9890 

•9894 

.9899 

•9903 

.9908 

°l 

2 

2 

3 

3 

4 

4 

98  .99" 
99  .9956 

.9917 
.9961 

.9921 
•9965 

.9926 
.9969 

•993° 
•9974 

•9934 
.9978 

•9939 
•9983 

•9943 
.9987 

.9948 
.9991 

•9952 
.9996 

3 

2 
2 

2 
2 

3 
3 

3 

4 
3 

4 

4 

44 


SURVEYING. 


TABLE    IIlA. 

LOGARITHMS  OF  SINES  AND  TANGENTS. 


o" 

i° 

Sin. 

Cos.   |  Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

o' 

o.oooo 

8.2419 

9.9999 

8.2419 

1.7581 

60' 

I 

6.4637 

.0000 

6.4637 

3  •  5363 

.2490 

•9999 

.2491 

.7509 

59 

2 

.7648 

.0000 

.7648 

.2352 

.2561 

•9999 

•  2562 

•  7438 

58 

3 

6  9408 

.0000 

6.9408 

3.0592 

.2630 

•9999 

•  2631 

•7369 

57  - 

4 

7-0658 

.0000 

7  0658 

2-9342 

.2699 

•9999 

.2700 

.7300 

56 

5 

.1627 

.0000 

..627 

•8373 

.2766 

•9999 

.2767 

•7233 

55 

6 

.2419 

.0000 

.2419 

.7581 

.2832 

•9999 

-2833 

.7167 

54 

7 

.3088 

.0000 

.3088 

.6912 

.2898 

•9999 

.2899 

.7101 

53 

8 

.3668 

.0000 

.3668 

•  6332 

.2962 

•9999 

.2963 

•7037 

52 

9 

.4.80 

.0000 

.4180 

.5820 

•3025 

•9999 

.3026 

.6974 

5l 

10 

•4637 

.0000 

•4637 

•5363 

.3088 

•9999 

.3089 

.6911 

50 

ii 

•SOS1 

.0000 

•5051 

•4949 

•  3T5° 

•9999 

•3*5° 

.6850 

49 

12 

•5429 

.0000 

•5429 

•4571 

.3210 

•9999 

.3211 

.6789 

48 

13 

•  ^777 

.0000 

•5777 

.4223 

.3270 

•9999 

•3271 

.6729 

47 

M 

.6099 

.OOOJ 

.6099 

.3901 

•3329 

•9999 

•3330 

.6670 

46 

15 

.6398 

.000  ) 

.6398 

.3602 

.3388 

•9999 

•3389 

.6611 

45 

16 

.6678 

.0000   !    .6678 

•  3322 

•3445 

•9999 

•3446 

•6554 

44 

17 

.6942 

.0000 

.6942 

•3058 

.3502 

•9999 

•35°3 

.6497 

43 

18 

.7190 

.0000 

.7190 

.2810 

•3558 

•9999 

•3559 

.644t 

42 

19 

•7425 

.0000 

•7425 

•2575 

•36*3 

•9999 

.3614 

.6386 

41 

20 

.7648 

.0000 

.7648 

•2352 

.3668 

•9999 

•  3669 

•6331 

40 

21 

•7859 

.0000 

.7860 

.2140 

.3722 

•9999 

•3723 

.6277 

39 

22 

.8061 

.0000 

.8062 

.1938 

•3775 

•  9999 

•3776 

.6224 

38 

23 

•8255 

.0000 

.8255 

•'745 

.3828 

•9999 

.3829 

.6171 

37 

24 

.8439 

.0000 

.8439 

•1561 

.3880 

•9999 

•  3881 

.6119 

36 

25 

.8617 

.0000 

.8617 

•3931 

•9999 

•3932 

.6068 

35 

26 

.8787 

.0000 

•8787 

.1213 

.3982 

•9999 

•3983 

.6017 

34 

27 

•895' 

.0000 

.8951 

.1049 

.4032 

•9999 

•  4033 

•5967 

33 

28 

.9109 

.0000 

.9109 

.0891 

.4082 

•9999 

•  4083 

•5917 

32 

29 

.9261 

.0000 

.9261 

.0739 

•4»3* 

•9999 

•4T32 

.5868 

31 

3° 

.9408 

.0000 

.9409 

.0591 

•4'79 

•9999 

.4181 

.5819 

3° 

31 

•9551 

.0000 

g-Si 

.0449 

.4227 

.9998 

.4229 

•5771 

29 

32 

.9689 

.0000 

.9689 

.0311 

•4275 

.9998 

.4276 

•5724 

28 

33 

.9822 

.0000 

•  9823 

.0177 

•4322 

•9998 

•4323 

•5677 

27 

34 

7-99S2 

.0000 

7.9952 

2.0048 

.4368 

.9998 

•437° 

.5630 

26 

35 

8.0078 

.0000 

8.0078 

1.9922 

.4414 

.9998 

.4416 

•5584 

25 

36 

.0200 

.0000 

.0200 

.9800 

•4459 

.9098 

.4461 

•5539 

24 

37 

.0319 

.0000 

.0319 

.9681 

•4504 

.9998 

.4506 

•5494 

23 

38 

•°435 

.0000 

•0435 

•9565 

•4549 

.9908 

•4551 

•5449 

22 

39 

.0548 

.0000 

.0548 

•9452 

•  4593 

.9998 

•4595 

•5405 

21 

40 

.0658 

.0000 

.0658 

•9342 

•4637 

.9998 

.4638 

•SS62 

20 

41 

.0765 

.0000 

.0765 

•9235 

.4680 

.9998 

.4682 

•5318 

I9 

42 

.0870 

.0000 

.0870 

.9130 

•4723 

.9998 

•4725 

•5275 

18 

43 

.0972 

.0000 

.0972 

.9028 

•4765 

.9998 

.4767 

•5233 

17 

44 

.1072 

.0000 

.1072 

.8928 

.4807 

.9998 

.4809 

•S'91 

16 

45 

.  1169 

.0000 

.1170 

.8830 

.4848 

.9998 

.4851 

•5H9 

15 

46 

.1265 

.0000 

.1265 

•8735 

.4890 

.9998 

.4892 

-5108 

14 

47 

•1358 

.0000 

•1359 

.8641 

.4930 

•9998 

•4933 

.5067 

13 

48 

.1450 

.0000 

.1450 

•  8550 

.4971 

.9998 

•4973 

.5027 

12 

49 

•1539 

.0000 

.1540 

.8460 

.5011 

.9998 

•5013 

•4987 

II 

5° 

.1627 

.0000 

.  1627 

•8373 

•5050 

.9998 

•5053 

•4947 

IO 

51 

•T7'3 

.0000 

•I7I3 

.8287 

.5090 

.9998 

.5092 

.4908 

9 

52 

•'797 

o.oooo 

•  1798 

.8202 

.5^29 

.9998 

•5I3I 

.4869 

8 

53 

.1880 

9  •  9999 

.1880 

.8120 

•  5^7 

.9998 

.5170 

•  4830 

7 

54 

.  1961 

•9999 

.1962 

.8038 

.5206 

•9998 

.5208 

•4792 

6 

55 

.2041 

•9999 

.2041 

•7959 

.5243 

.9998 

.5246 

•4754 

5 

56 

.2119 

•9999 

.2120 

.7880 

.5281 

.9998 

•5283 

.4-717 

4 

57 

.2196 

•  9999 

.2196 

.7804 

•53*8 

•9997 

•5321 

.4679 

3 

58 

.2271 

•9999 

.2272 

.7728 

•5355 

•9997 

•5358 

.4642 

2 

g 

.2346 
8.2419 

•9999 
9-9999 

.2346 
8.2419 

•7654 
i-758i 

•5392 
8.5428 

•9997 
9-9997 

•5394 
8.5431 

.4606 
1-4569 

I 
O 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

89" 

88° 

TABLES. 


45 


TABLE    \\\^.— Continued. 
LOGARITHMS  OF  SINES  AND  TANGENTS. 


2° 

3° 

4° 

Sin. 

Cos. 

Tan. 

Cot. 

Sin. 

Cos. 

Tan. 

Cot. 

Sin.  |  Cos. 

Tan. 

Cot. 

o' 

.5428 

9-9997 

•543* 

1.4569 

.7188 

9-9994 

8.7194 

.2806 

8.84369.9989 

8.8446 

I.IS54 

60' 

I 

•  5464 

•9997 

•5467 

-4533 

.7212 

•9994 

.7218 

.2782 

•  8454 

.9989  .8465 

•1535 

59 

2 

•  55°o 

•9997 

•5503 

•4497 

.7236 

•9994 

•  7242 

.2758 

•8472 

.9989 

.8483 

•1517 

58 

3 

•5535 

•9997 

•5538 

.4462 

.7260 

•9994 

.7266 

•  2734 

.8490!  .9989 

.8501 

.1499 

57 

4 

•5571 

•9997 

•5573 

.4427 

•7283 

•9994 

.7299 

.2710 

.8508!  .9989 

-8518 

.1482 

56 

5 

•5605 

•9997 

.5608 

•4392 

•7307 

•9994 

•7313 

.2687 

•  8525 

.9989 

.8536 

.1464 

55 

6 

.5640 

•9997 

•5643 

•4357 

•733° 

•9994 

•7337 

.2663 

•  8543 

•9989 

•8554 

.1446 

54 

7 

.5674 

•9997 

•5677 

•  4323 

•7354 

•9994 

.7360 

.2640 

.8560 

.9989 

•8572 

.1428 

53 

8 

.5708 

•9997 

.5711 

.4289 

•7377 

•9994 

•7383 

.2617 

.8578 

.9989 

•  8589 

.1411 

52 

9 

•5742 

•9997 

•5745 

•4255 

.7400 

•9993 

.7406 

•  2594 

•8595 

•9989 

.8607 

•1393 

5» 

10 

•5776 

•9997 

•5779 

.4221 

•7423 

•9993 

.7429 

•2571 

.8613 

•9989 

.8624 

•1376 

50 

ii 

.5809 

•  9997 

.5812 

.4188 

•7445 

•9993 

•7452 

.2548 

.8630 

.9988 

.8642 

.1358 

49 

12 

.5842 

•9997 

.5845 

•4^5 

.7468 

•9993 

•7475 

•  2525 

.8647 

.9988 

.8659 

.1341 

48 

13 

•5875 

•9997 

.5878 

.4122 

.7491 

•9993 

•7497 

•  250? 

•  8665 

.9988 

.8676 

•  1324 

47 

*4 

•5907 

•9997 

•5911 

.4089 

•7513 

•9993 

.7520 

.2480 

.8682 

.9988 

.8694 

.1306 

46 

i5 

•5939 

•9997 

•5943 

•4°57 

•7535 

•9993 

•7542 

.2458 

.8699 

.9988 

.8711 

.1289 

45 

16 

•5972 

•9997 

•5975 

.4025 

•7557 

•9993 

.7565 

•  2435 

.8716 

.9988 

.8728 

.1272 

44 

17 

.6003 

•9997 

.6007 

•3993 

.7580 

•9993 

•7587 

'24!3 

•8733 

.9988 

•8745 

•  1255 

43 

18 

•  6035 

.9996 

.6038 

.3962 

.7602 

•9993 

.7609 

.2391 

•8749 

.9988 

.8762 

.1238 

42 

*9 

.6066 

.9996 

.6070 

•  393° 

.7623 

•9993 

•763! 

.2369 

.8766 

.9988 

.8778 

.  1222 

41 

20 

.6097 

.9996 

.6101 

•3899 

•7645 

•9993 

•7652 

.2348 

.8783 

.9988 

•8795 

.1205 

43 

21 

.6!28 

.9996 

.6132 

.3868 

.7667 

•9993 

.7674 

.2326 

.8799 

•9987 

.8812 

.1188 

39 

22 

.6159 

.9996 

.6163 

.3837 

.7688 

.9992 

.7696 

.2304 

.8816 

•9987 

.8829 

.1171 

38 

23 

.6189 

•9996 

.6193 

.3807 

.7710 

.9992 

.7717 

.2280 

-8833 

.9987 

-8845 

•1155 

37 

24 

.6220 

.9996 

.6223 

•3777 

•7731 

.9992 

•7739 

.2261 

.8849 

•  9987 

.8862 

.1138 

36 

25 

.6250 

.9996 

.6254 

•3746 

•7752 

.9992 

.7760 

.2240 

.8865 

.9987 

.8878 

.1122 

35 

26 

.6279 

.9996 

.6283 

•3717 

•7773 

.9992 

.778i 

.2219 

.8882 

.9987 

.8895 

.1105 

34 

27 

.6309 

.9996 

•6313 

•3687 

-7794 

.9992 

.7802 

.2198 

.8898 

.9987 

.8911 

.1089 

33 

28 

•  6339 

.9996 

•^343 

•3657 

-7815 

.9992 

•7823 

.2177 

.8914 

9987 

.8927 

•1073 

32 

29 

.6368 

.9996 

.6372 

.3628 

.7836 

.9992 

.7844 

.2156 

.8930 

•9987 

•  8944 

.  1056 

31 

30 

•6397 

.9996 

.6401 

•  3599 

•7857 

•9992 

.7865 

•2135 

.8946 

.9987 

.8960 

.1040 

30 

31 

.6426 

.9996 

.6430 

•357° 

.7877 

.9992 

.7886 

.211, 

.8962 

.9986 

.8976 

.  IO2j( 

29 

32 

•6454 

.9996 

•6459 

•3541 

.7898 

•  9992 

.7906 

.2094 

.8978 

.9986 

.8992 

.1008 

28 

33 

.6483 

.9996 

.6487 

•3513 

.7918 

•9992 

.7927 

.2073 

.8994 

.9986 

.9008 

.0992 

27 

34 

.6511 

.9996 

•  6s?5 

-3485 

•7939 

•9992 

•7947 

.2053 

.9010 

.9986 

.9024 

.0976 

«26 

35 

•  6539 

.9996 

•6544 

.3456 

•7959 

.9992 

.7967 

.2033 

.9026 

.9986 

.9040 

.0960 

25 

36 

.6567 

.9996 

•6571 

•342y 

•7979 

.9991 

.7988 

.2OI2 

•9342 

.9986 

.9056 

.0944 

24 

37 

•6595 

•9995 

.6599 

.3401 

.7999 

.9991 

.8008 

.1992 

•9°57 

.9986 

.9071 

.og2C 

23 

38 

.6622 

•9995 

.6627 

•3373 

.8019 

.9991 

.8028 

.1972 

•9073 

.9986 

.9087 

.0913 

22 

39 

.6650 

•9995 

.6654 

•3346 

.8039 

.9991 

.8048 

•1952 

.9089 

.9986 

.9103 

.0897 

21 

4° 

.6677 

•9995 

.6682 

.3318 

.8059 

.9991 

.8067 

•193^ 

.9104 

.9986 

.9118 

.0882 

2O 

4* 

.6704 

•9995 

.6709 

.3291 

.8078 

.9991 

.8087 

.igi. 

.9119 

•9985 

•9i34 

.0866 

19 

42 
43 

•673! 

.6758 

•9995 
•9995 

.6736 
.6762 

•  3264 
.3238 

.8098 
.8117 

.9991 
.9991 

.8107 

'8l2( 

.1893 
.1874 

•9i35 
.9150 

•9985 
•  9985 

.9150 
.9165 

.0850 
.0835 

17 

44 

.6784 

•9995 

.6789 

.3211 

-8i37 

.9991 

.8146 

.1854 

.9166 

•9985 

.9180 

.0820 

16 

45 

.6810 

•9995 

.6815 

.3^85 

•  8156 

.9991 

•  8165 

.1835 

.9181 

•9985 

.9196 

.0804 

15 

46 

.6837 

•9995 

.6842 

•  3158 

•8i75 

.9991 

•  8185 

.181^ 

.9196 

.9985 

.9211 

.0789 

14 

47 

.6863 

•9995 

.6868 

•3T32 

.8194 

.9991 

.8204 

.1796 

.9211 

•9985 

.9226 

.0774 

'3 

48 

.6889 

•9995 

.6894 

.3106 

.8213 

.9990 

.8223 

.1777 

.9226 

•  9985 

.9241 

•0759 

12 

49 

.6914 

•9995 

.6920 

.3080 

.8232 

•999° 

.8242 

•!7S8 

.9241 

•9985 

.9256 

•0744 

II 

50 

.6940 

•9995 

.6945 

.3055 

-8251 

.9990 

.8261 

•1739 

.9256 

.9985 

.9272 

.072^ 

10 

5i 

.6965 

•9995 

.6971 

.3029 

.8270 

.9990 

.8280 

.1720 

.9271 

•  9984 

.9287 

.0713 

9 

52 

.6991 

•9995 

.6996 

.3004 

.8289 

.9990 

.8299 

.170 

.9286 

•9984 

.9302 

.0608 

8 

53 

.7016 

•9994 

.7021 

•2979 

.8307 

•  9990 

•8317 

.1683 

.9301 

•  9984 

.9316 

.0684 

7 

54 

.7041 

•9994 

.7046 

•2954 

.8326 

.9990 

.8336 

.1664 

•93*5 

•9984 

•9331 

.0669 

6 

55 

.7066 

•9994 

.7071 

.2929 

•8345 

.9990 

•8355 

.1645 

•9330 

•9984 

•9346 

.06S4 

5 

56 

.7090 

•9994 

.7096 

.2904 

.8363 

.9990 

•8373 

.1627 

•9345 

•  9984 

.9361 

.0639 

4 

57 

•7"5 

•9994 

.712 

.2879 

.8381 

.9990 

.8392 

.1608 

•9359 

.9984 

•  9376 

.0624 

3 

58 

.7140 

•9994 

•  7145 

.2855 

.8400 

•  99QQ 

.8410 

.1590 

•9374 

•  9984 

•939° 

.0610 

a 

59 

.7164 

•9994 

.7170 

•  2830 

.8418 

.9989 

.8428 

•I572 

.9388 

•9984 

•9405 

•°595 

i 

60 

8.7188 

9-9994 

8.7194 

i  .  2806 

8  8436 

9-9989 

8.8446 

I-I554 

8  9403 

9-9983 

8.9420 

1.0580 

0 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

Cos. 

Sin. 

Cot. 

Tan. 

87° 

86° 

85° 

46 


SUR  VE  YING. 


TABLE    II  lA— Continued. 
LOGARITHMS  OF  SINES  AND  TANGENTS. 


Arc.  Sin.  jDf.|  Cos. 

Df.  Tan.  IDf.  Cot. 

Arc. 

Arc. 

Sin. 

Df. 

Con. 

Df.  Tan.  Df. 

Cot. 

Arc. 

0   1 

0  / 

0  / 

0  / 

5  c 

IO 

8.9403 
•9545 

142 

9.9983 
.9982 

8.9420 
•9563 

143 

138 

1.0580 
•0437 

85  o 
5° 

10 

•4'77 

47 
46 

9.9849 
.9846 

3 

3 

9.4281 
•4331 

So 
50 

°-57i9 
.5669 

75  o 

5° 

20 

.9682 

T4 

.9981 

.9701 

T35 

.0299 

40 

20 

.4223 

46 

•9843 

4 

.4381 

49 

•  5619 

40 

30 

.9816 

129 

.9980 

•9836 

130 

.0164 

3° 

30 

.4269 

45 

•9839 

3 

•443° 

49 

•5570 

3° 

40 

8-9945 

125 

•9979 

8.9966 

127  1.0034 

20 

40 

•43M 

45 

.9836 

4 

•4479 

48 

•552i 

20 

50 

9.0070 

122 

•9977 

9.0093 

1230.9907 

10 

50 

•4359 

44 

•9832 

4 

•4S27 

48 

•5473 

10 

60 

.0192 

1  10 

.9976 

.0216 

I2O 

.9784 

840 

16  o 

•4403 

44 

.9828 

3 

•4575 

47 

•5425 

74  o 

10 

.0311 

"5 

•9975 

•0336 

117 

.9664 

50 

IO 

•4447 

44 

.9825 

4 

.4622 

47 

•5378 

5° 

20 

.0426 

"3 

•9973 

•0453 

•9547 

40 

20 

.4491 

42 

.9821 

4 

.4669 

47 

•5331 

40 

30 

.0539 

109 

.9972 

.0567 

III 

•9431 

30 

3° 

•4533 

43 

.9817 

3 

.4716 

46 

.5284 

3° 

40 

.0648 

107 

.9971 

.0678 

108 

•9322 

20 

40 

•4576 

42 

.9814 

4 

.4762 

46 

20 

50 

•0755 

104 

.9969 

.0786 

105 

.9214 

IO 

So 

.4618 

41 

.9810 

4 

.4808 

45 

•5192 

10 

7  o 

.0859 

102 

.9968 

.0891 

104 

.9109 

830 

17  o 

.4659 

41 

.9806 

4 

•4853 

45 

•5I47 

73  o 

10 

.0961 

99 

.9966 

•0995 

101 

.9005 

50 

10 

.4700 

41 

.9802 

4 

.4898 

45 

.5102 

5° 

20 

.  1060 

97 

.9964 

.  1096 

98 

.8904 

40 

20 

•4741 

40 

.9798 

4 

•4943 

44 

.5057 

40 

30 

•«57 

95 

•9963 

•"94 

97 

.8806 

30 

30 

.4781 

4O 

•9794 

4 

•4987 

44 

.5013 

3° 

40 

-1252 

93 

.9961 

.  1291 

94 

.8709 

20 

40 

.4821 

49 

.9790 

4 

•5031 

44 

.4969 

20 

50 

•1345 

•9959 

•1385 

93 

.8615 

IO 

50 

.4861 

39 

.9786 

4 

•5°75 

43 

•4925 

IO 

8  o 

•1436 

89 

.9958 

-I478 

01 

.8522 

82  o 

18  o 

.4900 

39 

.9782 

4 

.5118 

43 

.4882 

72  o 

IO 

.1525 

•9956 

.1569 

8') 

.8431 

5° 

IO 

•4939 

38 

.9778 

4 

.5161 

42 

•4839 

5° 

20 

.  1612 

85 

•9954 

.1658 

87 

.8342 

40 

20 

•4977 

38 

•9774 

4 

•5203 

42 

•4797 

40 

30 

.1697 

84 

•9952 

•1745 

86 

•8255 

30 

30 

•  50'  5 

37 

.9770 

5 

•5245 

4* 

•4755 

3° 

40 

.178! 

82 

.9950 

•  1831 

84 

8169 

20 

40 

•  SOS2 

38 

•9765 

4 

.5287 

42 

•4713 

20 

50 

•  1863 

80 

•9948 

.1915 

82 

.8085 

10 

50 

.5090 

36 

.9761 

4 

•5329 

41 

.4671 

10 

9  o 

•T9O 

79 

.9946 

.1997 

Bi 

.8003 

81  o 

19  o 

.5126 

37 

•9757 

5 

•537° 

41 

.4630 

71  o 

10 

.2022 

78 

•9944 

.20;  j 

So 

.7922 

So 

10 

.5153 

•9752 

4  ' 

•54" 

40 

•4589 

50 

20 

.2100 

76 

.9942 

.2158 

78 

.7842 

40 

20 

.5199 

36 

.9748 

S 

•5451 

40 

•4549 

40 

30 

.2176 

75 

•9940 

•  2236 

77 

.7764 

30 

30 

•5235 

3c; 

•9743 

4 

•5491 

40 

•4509 

30 

40 

•  2251 

73 

•9938; 

.2313 

76 

•  7687 

20 

40 

.5270 

36 

•9739 

5 

•5531 

4° 

.4469 

20 

50 

•2324 

73 

•9936 

.2389 

74 

.7611 

10 

5° 

.5306 

35 

•9734 

4 

•5571 

40 

•4429 

IO 

IO  O 

.2397 

71 

•  9934 

.2463 

73 

•  7537 

80  o 

20  0 

•5341 

34 

•9730 

S 

.5611 

39 

.4389 

70  o 

10 

.2468 

7° 

•9931 

.2536 

73 

.7464 

5° 

IO 

•5375 

34 

•9725 

4 

.5650 

39 

•435° 

5° 

20 

•2538 

63 

.9929 

.2609 

71 

•7391 

40 

20 

•5409 

34 

.9721 

5 

.5689 

38 

•43" 

40 

30 

.2606 

68 

•9927  3 

.2680 

70 

•7320 

30 

3° 

•5443 

34 

.9716 

.5 

•5727 

39 

•4273 

3° 

4° 

.2674 

66 

•9924   2 

•  2750 

69 

.7250 

2O 

40 

•5477 

33 

•97" 

5 

.5766 

38 

•4234 

20 

5° 

.2740 

66 

.9922 

3 

.2819 

68 

.7181 

10 

50 

•5510 

33 

.9706 

4 

.5804 

33 

.4196 

IO 

II  0 

.2806 

64 

.9919 

2 

.2887 

66 

.7113 

79  o 

21  0 

-5543 

33 

.9702 

5 

.5842 

37 

•4158 

69  o 

IO 

.2870 

64 

.9917 

3 

•2953 

67 

.7047 

50 

10 

.5576 

33 

.9697 

5 

•5879 

38 

.4121 

5° 

20 

•2934 

63 

.9914 

2 

.3020 

65 

.6980 

40 

2O 

.5609 

32 

.9692 

5 

•5917 

37 

.4083 

40 

30 

.2997 

61 

.9912 

3 

•3085 

64 

.6915 

30 

30 

.5641 

S2 

.9687 

5 

•5954 

37 

.4046 

3° 

40 

•3058 

61 

.9909 

2 

•3H9 

63 

.6851 

20 

40 

•  5673 

31 

.9682 

5 

37 

.4009 

20 

5° 

•3"9 

60 

.9907 

3 

•  3212 

63 

.6788 

IO 

50 

•57°4 

32 

.9677 

5 

^6028 

36 

•3972 

10 

12  O 

•3*79 

59 

•  99°4 

3 

•3275 

6l 

•  6725 

78  o 

22  0 

•5736 

31 

.9672 

5 

.6064 

36 

•3936 

68  o 

10 

•3238 

58 

.Q^OIj  2 

•3336 

61 

.6664 

5° 

IO 

•5767 

3l 

.9667 

6 

.6100 

36 

.3900 

50 

20 

.3296 

57 

.9899  3 

•3397 

61 

.6603 

40 

20 

•5798 

30 

.9661 

5 

.6136 

36 

•  3864 

40 

30 

•3353 

57 

.9896,  3 

•3458 

59 

.6542 

3° 

3° 

.5828 

31 

•  9656 

8 

.6172 

.,f 

.3828 

3° 

40 

.3410  56 

•9893'  3 

•35r7 

59 

•  6483 

20 

40 

•5859 

.3° 

.9651 

5 

.6208 

3S 

•3792 

20 

5° 

-3466 

55 

.9890 

3 

•3576 

58 

•  6424 

IO 

So 

.5889 

30 

.9646 

6 

.6243 

36 

•3757 

IO 

'3  o 

•3521 

54 

.9887 

3 

•3634 

57 

.6366 

77  o 

23  o 

•59*9 

29 

.9640 

5 

.6279 

35 

•3721 

67  o 

IO 

•3575 

54 

•988*  3 

.3691 

57 

.6309 

50 

10 

•5948 

3° 

.9635 

6 

•63M 

34 

.3686 

50 

20 

.3629 

53 

.9881;  3 

•3748 

56 

.6252 

40 

20 

•5978 

•9 

.9629 

5 

.6348 

35 

•3652 

40 

30 

.3682 

52 

.9878'  3 

.3804 

55 

.6196 

30 

30 

.6007 

29 

.9624 

6 

.6383 

34 

•  36l7 

3° 

40 

•3734 

52 

•9875!  3 

•3859 

55 

.614. 

20 

40 

.6036 

29 

.9618 

m 

.6417 

35 

.3583 

20 

50 

.3786 

.9872  3 

•39*4 

54 

.6086 

10 

50 

.6065 

28 

•  96*3 

6 

.6452 

34 

•3548 

IO 

14  o 

•3837 

5° 

.9869  3 

•  3968 

53 

.6032 

76  o 

24  o 

.6093 

28 

.9607 

5 

.6486 

34 

•35*4 

66  o 

10 

.3887 

So 

.9866  3 

.4021 

53 

•5979 

50 

10 

.6121 

28 

.9602 

6 

.6520 

33 

.3480 

5° 

20 

•3937 

49 

.9863  4 

•4°74 

53 

.5926 

40 

20 

.6149 

28 

•9596 

6 

•6553 

34 

•3447 

40 

30 

.3986 

49 

•9859  3 

.4127 

52 

•5873 

30 

30 

.6177 

28 

•9590 

6 

.6587 

33 

•34i3 

30 

40 

.4035 

48 

.9856  3 

.4178 

-5822 

20 

40 

.6205 

27 

.9584 

5 

.6620 

34 

20 

50 

.4083 

47 

•9853  4 

.4230 

5i 

•577° 

IO 

5° 

.6232 

27 

•9579 

6 

.6654 

33 

•3346 

IO 

'5  o 

9-4I30 

47 

9-9849  3 

9.4281 

500.5719 

75  o 

25  o 

0.6259 

27 

9-9573 

7 

9.6687 

33 

o-33l3 

650 

Arc. 

COB. 

Df. 

Sin.  Df. 

Cot. 

Df.  Tan. 

Arc. 

Arc. 

Cos. 

Df. 

Sin. 

Df. 

Cot. 

Df. 

Tan. 

Arc. 

TABLES. 


47 


TABLE   IIU— Continued. 
LOGARITHMS  OF  SINES  AND  TANGENTS. 


Arc. 

Sin. 

->f. 

Cos. 

Df. 

Tan. 

Df. 

Cot. 

Arc. 

Arc. 

Sin. 

Df. 

Cos. 

Df. 

Tan.  |Df. 

Cot. 

Arc. 

0  / 

0  / 

0  / 

0  / 

25  o 

9.6259 

27 

9-9573 

6 

9.6687 

33 

0.3313 

65  o 

35  o 

9.7586 

18 

9-9*34 

9 

9.8452 

27 

0.1548 

55  o 

10 

.6286 

27 

.9567 

6 

.6720 

32 

.3280 

50 

IO 

.7604 

18 

•9I25 

9 

.8479 

27 

.1521 

5° 

20 

•6313 

27 

.9561 

6 

.6752 

33 

.3248 

40 

20 

.7622 

18 

.9116 

9 

.8506 

27 

.1494 

40 

30 

.6340 

26 

•9555 

6 

.6785 

32 

•  3215 

30 

3° 

.7640 

i7 

.9107 

9 

•8533 

26 

.1467 

3° 

40 

.6366 

26 

•9549 

6 

.6817 

33 

•3lS3 

20 

40 

•7657 

18 

.9098 

9 

•8559 

27 

.1441 

20 

50 

.6392 

26 

•9543 

6 

.6850 

32 

•3*50 

IO 

50 

•7675 

17 

.9089 

9 

.8586 

27 

.1414 

10 

26  o 

.6418 

26 

•  9537 

7 

.6882 

32 

.3118 

64  o 

36  o 

.7692 

18 

.9080 

IO 

.8613 

26 

•1387 

54  o 

IO 

.6444 

26 

•953° 

6 

.6914 

32 

.3086 

5° 

IO 

.7710 

jy 

.9070 

9 

.8639 

27 

.1361 

50 

20 

.6470 

25 

•9524 

6 

.6946 

.3054 

40 

20 

.7727 

J7 

.9061 

9 

.8666 

26 

•1334 

40 

3° 

.6495 

26 

.9518 

6 

.6977 

S2 

•3023 

30 

30 

•7744 

17 

.9052 

IO 

.8692 

26 

.1308 

30 

40 

.6521 

25 

.9512 

7 

.7009 

.2991 

20 

40 

.7761 

17 

.9042 

9 

.8718 

27 

.1282 

20 

50 

.6546 

24 

•95°5 

6 

.7040 

32 

.2960 

IO 

5° 

.7778 

17 

•9033 

10 

•8745 

26 

'1255 

10 

27  o 

.6570 

25 

•9499 

7 

.7072 

31 

.2928 

63  o 

37  o 

•7795 

16 

•  9023 

9 

.8771 

26 

.1229 

53  ° 

10 

•6595 

25 

.9492  6 

•  7103 

.2897 

5° 

IO 

.7811 

17 

.9014 

IO 

•8797 

27 

.1203 

5° 

20 

.6620 

24 

.9486 

7 

•7134 

31 

.2866 

40 

20 

.7828 

16 

.9004 

9 

.8824 

26 

.1176 

40 

3° 

.6644 

24 

•9479 

6 

•  7l65 

31 

•2835 

3° 

3° 

.7844 

17 

•8995 

IO 

.8850 

26 

.1150 

3° 

4° 

.6668 

•9473 

7 

.7196 

30 

,2804 

20 

4° 

.7861 

16 

.8985 

IO 

.8876 

26 

.1124 

20 

50 

.6692 

24 

.9466 

7 

.7226 

3' 

•2774 

IO 

5° 

.7877 

1  6 

•8975 

10 

.8902 

26 

.1098 

10 

28  o 

.6716 

24 

•94S9 

6 

•7257 

3° 

•2743 

6a  o 

38  o 

.7893 

17 

.8965 

IO 

.8928 

26 

.1072 

52  o 

1.0 

.6740 

23 

•9453 

7 

.7287 

3° 

•27I3 

5° 

IO 

.7910 

16 

•8955 

IO 

•  8954 

26 

.  1046 

5° 

20 

.6763 

24 

.9446 

7 

•7317 

3* 

.2683 

40 

20 

.7926 

15 

•8945 

10 

.8980 

26 

.1020 

40 

3° 

.6787 

23 

•  9439 

7 

•7348 

3" 

.2652 

3° 

3° 

.7941 

16 

•8935 

to 

.9006 

26 

.0994 

3° 

40 

.6810 

23 

.9432 

7 

•7378 

30 

.2622 

20 

40 

•7957 

16 

.8925 

xo 

.9032 

af> 

.0968 

20 

50 

-6833 

23 

•9425 

7 

.7408 

3° 

.2592 

10 

5° 

•7973 

16 

.8915 

10 

.9058 

26 

.0942 

10 

29  o 

.6856 

22 

.9418 

7 

.7438 

29 

.2562 

61  o 

39  o 

.7989 

15 

.8905 

10 

.9084 

26 

.091^ 

51  o 

10 

.6878 

23 

.9411 

7 

.7467 

3° 

•2533 

50 

IO 

.8004 

16 

.8895 

XI 

.9110 

25 

.0890 

50 

20 

.690! 

22 

.9404 

7 

•7497 

29 

•2503 

40 

20 

.8020 

15 

.8884 

IO 

•9*35 

2fc 

.0865 

40 

3° 

•  6923 

23 

•9397 

7 

.7526 

3° 

.2474 

3° 

3° 

•  8035 

15 

.8874 

10 

.9161 

26 

.0839 

3° 

40 

.6946 

22 

•939° 

7 

•7556 

29 

.2444 

20 

4P 

•  8050 

1  6 

.8864 

ii 

.9187 

25 

.0813 

20 

50 

.6968 

22 

•9383 

8 

•7585 

29 

•2415 

TO 

5° 

.8066 

15 

-8853 

10 

.9212 

26 

.0788 

10 

30  o 

.6990 

22 

•9375 

7 

.7614 

3° 

.2386 

60  o 

40  o 

.8081 

15 

.8843 

II 

.9238 

26 

.0762 

50  0 

IO 

.7012 

21 

.9368 

7 

.7644 

29 

•2356 

5° 

IO 

.8096 

15 

.8832 

II 

.9264 

25 

.0736 

5° 

20 

•7033 

22 

.9361 

8 

•7673 

28 

.2327 

40 

20 

.8111 

M 

.8821 

II 

.9289 

26 

.0711 

40 

30 

•7055 

21 

•9353 

7 

.7701 

29 

.2299 

3° 

3° 

.8125 

15 

.8810 

IO 

•93^5 

26 

.0685 

3° 

40 

.7076 

21 

•9346 

8 

•773° 

29 

.2270 

20 

40 

.8140 

15 

.8800 

11 

•934' 

25 

.0659 

20 

50 

.7097 

21 

•9338 

7 

•7759 

29 

.2241 

10 

5° 

M 

.8789 

I! 

.9366 

26 

.0634 

10 

31  o 

.7118 

21 

•9331 

8 

.7788 

28 

.2212 

59  o 

41  o 

.8169 

15 

•  8778 

II 

•9392 

2S 

.0608 

49  ° 

10 

•7'39 

•21 

•9323 

8 

.7816 

29 

.2184 

5° 

10 

.8184 

14 

.8767 

11 

.941? 

26 

.0583 

50 

20 

.7160 

21 

•9315 

7 

.7845 

28 

•2155 

40 

20 

.8.98 

15 

.8756 

II 

•9443 

25 

•°557 

40 

30 

.7181 

90 

.9308 

8 

•7873 

29 

.2127 

3° 

30 

.8213 

T4 

•8745 

12 

.9468 

26 

•0532 

3° 

4° 

.7201 

21 

.9300 

8 

.7902 

2H 

.2098 

20 

40 

.8227 

J4 

•8733 

II 

•  9494 

25 

.0506 

20 

5° 

.7222 

2O 

.9292 

8 

•7930 

23 

.2070 

IO 

5° 

.8241 

.8722 

XX 

25 

.0481 

IO 

32  o 

10 

.7242 
.7262 

20 
20 

.9284 

.9276 

8 
8 

•7958 
.7986 

28 
28 

.2042 
.2014 

58  o 
5° 

42  o 

IO 

•8255 
.8269 

M 

M 

.8711 
.8699 

12 
11 

•9544 
•957° 

26 

25 

.0456 
.0430 

480 

5° 

20 

.7282 

2O 

.9268 

8 

.8014 

28 

.1986 

4° 

20 

.8283 

M 

.8688  12 

•9595 

26 

.0405 

40 

30 

.7302 

2O 

.9260 

8 

.8042 

28 

.1958 

30 

3° 

.8297 

M 

.8676  ii 

.9621 

25 

.0379 

3° 

40 

.7322 

2O 

.9252 

8 

.8070 

27 

.1930 

20 

40 

.8311 

13 

.8665   12 

.9646 

25 

•0354 

20 

5° 

•7342 

T9 

.9244 

8 

.8097 

28 

.1903 

IO 

50 

•8324 

14 

.8653!  I2 

.9671 

26 

.0329 

IO 

33  o 

.7361 

19 

.9236 

8 

.8125 

28 

.1875 

57  o 

43  o 

•8338 

13 

.8641)  12 

.9697 

25 

.0303 

47  o 

IO 

.7380 

20 

.9228 

9 

•  8153 

87 

.1847 

50 

IO 

•8351 

14 

.8629   II 

.9722 

25 

.0278 

5° 

•  20 

.7400 

19 

.9219 

8 

.8180 

28 

.1820 

40 

20 

•8365 

.8618 

12 

•9747 

25 

.0253 

40 

30 

.7419 

'9 

.9211 

8 

.8208 

27 

.1792 

3° 

3° 

.8378 

13 

.8606 

12 

.9772 

26 

.0228 

3° 

40 

.7438 

1C 

.9203 

9 

•  82^5 

28 

•1765 

20 

40 

.8391 

•8594 

12 

.9798 

25 

.0202 

20 

50 

•7457 

K 

.9194 

8 

.8263 

27 

IO 

50 

.8405 

13 

.8582 

13 

.9823 

25 

.0177 

IO 

34  o 

.7476 

18 

.9186 

9 

.8390 

27 

.1710 

560 

44  o 

.8418 

13 

.8569 

12 

.9848 

26 

.0152 

46  o 

10 

•7494 

ly 

.9177 

8 

.83.7 

27 

.1683 

5° 

IO 

-8431 

I3 

•8557 

12 

.9874 

25 

.OI26 

5° 

20 

•7513 

18 

.9169 

9 

•8344 

27 

-1656 

40 

20 

.8444 

13 

.8545 

13 

.9899 

25 

.0101 

40 

30 

•7531 

19 

.9160 

9 

•8371 

27 

.1629 

30 

30 

•8457 

12 

.8532 

12 

•9924 

M 

.0076 

3° 

4° 

•755C 

ii 

•  9151 

9 

•  8398 

27 

-I602 

20 

40 

.8469 

X3 

.8520 

X3 

•9949 

26 

.0051 

20 

50 

.7568 

18 

.9142 

8 

.8425 

27 

•1575 

IO 

5° 

.8482 

'3 

.8507 

12 

9-9975 

25 

.0025 

10 

35  o 

9.7586 

18 

9-9I34 

9 

9.8452 

27 

0.1548 

55  ° 

45  OJ9.8495 

9-8495 

o.oooo 

0.0000 

45  o 

Arc 

Cos. 

Df 

Sin. 

Df 

Cot. 

Df 

Tan. 

Arc. 

ArcJ  Cos. 

Df. 

Sin.  'Df. 

Cot. 

Df. 

Tan. 

Arc. 

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TABLES. 


53 


i 


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54 


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TABLES. 


55 


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'         1 


SURVEYING. 


TABLE   V. 
HORIZONTAL  DISTANCES  AND  ELEVATIONS  FROM  STADIA  READINGS.     §  204. 


| 

i 

0° 

1° 

2° 

3° 

Minutes. 

Hor. 

DifT. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

DifT. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

o     .     . 

100.00 

o.oo 

99-97 

1.74 

99.88 

3-49 

99-73 

5-23 

2       .       . 

" 

0.06 

" 

1.  80 

99.87 

3-55 

99.72 

5.28 

4     •     • 

" 

O.I2 

« 

1.86 

u 

3-6o 

99.71 

5-34 

6    .     . 

« 

O.I7 

99.96 

1.92 

«   • 

3-66 

u 

5-40 

8    .    . 

«« 

0.23 

u 

1.98 

99.86 

3-72 

99.70 

5-46 

10      .      . 

n 

O.29 

" 

2.04 

u 

3-/8 

99.69 

5.52 

12      .      . 

« 

o-35 

« 

2.09 

99-85 

3-84 

« 

5-57 

14      .      . 

« 

0.41 

99-95 

2.15 

it 

3-90 

99-68 

5-63 

16    .    . 

it 

0.47 

" 

2.21 

99.84 

3-95 

« 

5-69 

18    .    . 

« 

0.52 

" 

2.27 

" 

4.01 

99.67 

5-75 

20      .      . 

« 

0.58 

« 

2-33 

99-83 

4.07 

99.66 

5.80 

22      .      . 

« 

0.64 

99-94 

2.38 

« 

4-13 

« 

5.86 

24      .      . 

« 

0.70 

H 

2-44 

99.82 

4.18 

99.65 

5-92 

26      .      . 

99-99 

0.76 

(( 

2.50 

« 

4.24 

99.64 

5.98 

28      .      . 

(4 

o.Si 

99-93 

2.56 

99.81 

4-3° 

99-63 

6.04 

30      .      . 

« 

0.87 

« 

2.62 

« 

4-36 

" 

6.09 

32      .      . 

« 

o-93 

« 

2.67 

99.80 

4.42 

99.62 

6.15 

34    •    • 

" 

0.99 

« 

2-73 

" 

4.48 

M 

6.21 

36    •    • 

« 

1.05 

99.92 

2-79- 

99-79 

4-53 

99.61 

6.27 

38    •    • 

« 

i.  ii 

u 

2.85 

« 

4-59 

99.60 

6-33 

40    .    . 

" 

1.16 

" 

2.91 

99.78 

4-65 

99-59 

6.38 

42    .    . 

« 

1.22 

99.91 

2.97 

" 

4.71 

" 

6.44 

44     .    • 

99.98 

1.28 

". 

3.02 

99-77 

4.76 

99.58 

6.50 

46    .     . 

(( 

i-34 

99.90 

3-o8 

<( 

4.82 

99-57 

6.56 

48     .     . 

« 

1.40 

« 

3-1* 

99.76 

4.88 

99-56 

6.61 

50     .     . 

(( 

i-45 

" 

3.20 

« 

4-94 

" 

6.67 

52     •    • 

« 

i-5i 

99.89 

3.26 

99-75 

4-99 

99-55 

6-73 

54    •     • 

« 

'•57 

" 

3-3i 

99-74 

5-°5 

99-54 

6.78 

56     .     . 

99-97 

1.63 

<( 

3-37 

" 

5-11 

99-53 

6.84 

58     .     . 

" 

1.69 

99.88 

3-43 

99-73 

5-!7 

99-52 

6.90 

60    .    . 

« 

1.74 

" 

3-49 

" 

5-23 

99-51 

6.96 

'  =  o-75 

0.75 

O.OI 

0-75 

O.O2 

0.75 

0.03 

0-75 

0.05 

C=  I.OO 

I.OO 

O.OI 

I.OO 

O.O3 

I.OO 

0.04 

I.OO 

0.06 

r=i.25 

125 

O.O2 

1-25 

0.03 

1.25 

0.05 

1.25 

0.08 

^  *  This  table  was  computed  by  Mr.  Arthur  Winslow  of  the  State  Geological  Survey  of  Pennsylvania. 
For  description  of  chart  for  graphical  reduction  see  p.  v. 


TABLES. 


57 


TABLE  V. — Continued. 
HORIZONTAL  DISTANCES  AND  ELEVATIONS  FROM'  STADIA  READINGS. 


40 

5° 

6° 

70 

Minutes. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

O     .      . 

99.51 

6.96 

99-24 

8.68 

98.91 

10.40 

98.51 

12.  IO 

2      .      . 

7.02 

99-23 

8.74 

98.90 

10.45 

98.50 

12.15 

4     •     • 

99-5° 

7.07 

99.22 

8.80 

98.88 

10.51 

98.48 

12.21 

6    .    . 

99-49 

7-13 

99.21 

8.85 

98.87 

ro-57 

98.47 

1226 

8    .    . 

99.48 

7.19 

99.20 

8.91 

98.86 

10.62 

98.46 

12.32 

10    .     . 

99-47 

7-25 

99.19 

8.97 

98.85 

10.68 

98.44 

12.38 

12      .      . 

99.46 

7-30 

99.18 

9-°3 

98.83 

10.74 

98.43 

12.43 

14      .       . 

" 

7-36 

99.17 

9.08 

98.82 

10.79 

98.41 

12.49 

16    .    . 

99-45 

7-42 

99.16 

9.14 

98.81 

10.85 

98.40 

I2-55 

18    .    . 

99-44 

7.48 

99-15 

9.20 

98.80 

10.91 

98.39 

12.60 

20    .    . 

99-43 

7-53 

99.14 

9.25 

98.78 

10.96 

98.37 

12.66 

22      .      . 

•99.42 

7-59 

99-13 

9-31 

98.77 

1  1.  02 

98.36 

12.72 

24      .      . 

99.41 

7-65 

99.11 

9-37 

98.76 

11.08 

98.34 

12.77 

26  -.      . 

9940 

7.71 

99.10 

9-43 

98.74 

11.13 

98.33 

12.83 

28      .      . 

99-39 

7.76 

99.09 

9.48 

98.73 

11.19 

98.31 

12.88 

30      .      . 

9938 

7.82 

99.08 

9-54 

98.72 

11.25 

98.29 

12.94 

32      •      - 

99-38 

7.88 

99.07 

9.60 

98.71 

11.30 

98.28 

13.00 

34    •    • 

99-37 

7-94 

99.06 

9.65 

98.69 

11.36 

98.27 

T3-o5 

36    •    • 

99-36 

7-99 

99-05 

9.71 

98.68 

11.42 

98.25 

13.11 

38    •    • 

99-35 

8,05 

99.04 

9-77 

98.67 

11.47 

98.24 

i3-!7 

40    .    . 

99-34 

8.11 

99-03 

9-83 

98.65 

"•53 

98.22 

13.22 

42    .    . 

99-33 

8.17 

99.01 

9.88 

98.64 

11.59 

98.20 

13.28 

44    .    . 

99-32 

8.22 

9900 

9-94 

98-63 

11.64 

98.19 

J3-33 

46    .. 

99-  3  1 

8.28 

98.99 

10.00 

98.61 

11.70 

98.17 

13-39 

48     .     . 

99-30 

8-34 

98.98 

10.05 

98.60 

11.76 

98.16 

13-45 

50    .     . 

99.29 

8.40 

98.97 

IO.II 

98.58 

11.81 

98.14 

!3-5o 

52     .    . 

99.28 

8-45 

98.96 

10.17 

98.57 

11.87 

98.13 

13-56 

54    •    • 

99.27 

8.5I 

98.94 

10.22 

98.56 

11  -93 

98.11 

13.61 

56    .     . 

99.26 

8.57 

98.93 

10.28 

98.54 

11.98 

98.10 

13-67 

58     .     . 

99-25 

8.63 

98.92 

10.34 

98.53 

12.04 

98.08 

13-73 

60    .    . 

99.24 

8.68 

98.91 

10.40 

98.51 

12.10 

98.06 

13-78 

'  =  0.75 

0-75 

0.06 

0.75 

O.O7 

0-75 

o.oS 

0.74 

O.IO 

c  —  i.oo 

I.OO 

0.08 

0.99 

0.09 

0.99 

O.I  I 

0.99 

0.13 

'=1.25 

1.25 

O.IO 

1.24 

O.I  I 

1.24 

0.14 

1.24 

0.16 

SUR  VE  YING. 


TABLE  V.— Continued. 
HORIZONTAL  DISIANCES  AND  ELEVATIONS  FROM  STADIA  READINGS. 


8° 

9° 

1O° 

11° 

j 

Minutes. 

Hor. 

Dift 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

0      .      . 

98.06 

I3-78 

97-55 

«ws 

96.98 

17.10 

96.36 

18.73 

2       .      . 

98.05 

13.84 

97-53 

15-51 

96.96 

17.16 

96.34 

18.78 

4     •     • 

98.03 

13.89 

97-52 

15-56 

96.94 

17.21 

96.32 

18.84 

6    .     . 

98.01 

13-95 

97-5° 

15.62 

96.92 

17.26 

96.29 

l8.89 

8    .     . 

98.00 

I4.OI 

97.48 

15-67 

96.90 

I7.32 

96.27 

18.95 

10      .       . 

97.98 

14.06 

97.46 

15-73 

96.88 

17-37 

96.25 

19.00 

12      .      . 

97-97 

14.12 

97-44 

15-78 

96.86 

17-43 

96.23 

19.05 

I4      .      . 

97-95 

14.17 

97-43 

15.84 

96.84 

17.48 

96.21 

19.11 

16    .    . 

97-93 

14.23 

97  41 

15.89 

96.82 

17-54 

96.18 

19.16 

18    ..  . 

97.92 

14-28 

97-39 

15-95 

96.80 

T7-59 

96.16 

19.21 

20    .    . 

97.90 

14-34 

97-37 

16.00 

96.78 

17-65 

96.14 

19.27 

22 

97.88 

14.40 

"97-35 

1  6.06 

96.76 

17.70 

96.12 

19.32 

24 

97.87 

14-45 

97-33 

i6.n 

96.74 

17.76 

96.09 

19.38 

26      .      . 

97-85 

I4-51 

97-31 

16.17 

96.72 

17.81 

96.07 

19-43 

28      .      . 

97-83 

14.56 

97.29 

16.22 

96.70 

17.86 

96.05 

19.48 

30      .      . 

97.82 

14.62 

97.28 

16.28 

96.68 

17.92 

96.03 

19.54 

32      .      . 

97.80 

14.67 

97.26 

16.33 

96.66 

17.97 

96.00 

19-59 

34    •    • 

97-78 

M-73 

97.24 

16.39 

96.64 

18.03 

95-98 

19.64 

36    .. 

97.76 

14.79 

97-22 

16.44 

96.62 

18.08 

95-96 

19.70 

38     .     . 

97-75 

14.84 

97.20 

16.50 

96.60 

18.14  j     95.93 

J9-75 

40    .    . 

97-73 

14.90 

97.18 

16.55 

96.57 

18.19 

95-91 

19.80 

42    .     . 

97-71 

14-95 

97.16 

1  6.6  1 

96.55 

18.24 

95-89 

19.86 

44     •     . 

97.69 

15.01 

97.14 

1  6.66 

96.53 

i8'.3o 

95.86 

19.91 

46    .     . 

97.68 

15.06 

97.12 

16.72 

96.51 

18-35 

95.84 

19.96 

48     .     . 

97.66 

15.12 

97-iQ 

16.77 

96.49 

18.41 

95.82 

20.02 

50     .     . 

97.64 

I5-X7 

97.08 

16.83 

96.47 

18.46 

95-79 

20.07 

52     .     . 

97.62 

15-23 

97.06 

1  6.88 

96.45 

18.51 

95-77 

20.  1  2 

54     •     . 

97.61 

15.28 

97.04 

16.94 

96.42 

18.57 

95-75 

20.  1  8 

56    .     . 

97-59 

'5-34 

97.02 

16.99 

96.40 

18.62 

95-72 

20.23 

58     .     . 

97-57 

15.40 

97.00 

17.05 

96.38 

1  8.68 

95-70 

20.28 

60    .     . 

97-55 

*5-45 

96.98 

17.10 

96.36 

18.73 

95.68 

20-34 

'  =  0.75 

0.74 

O.I  I 

0.74 

0.12 

0.74 

0.14 

0-73 

0.15 

c—  i.oo 

0.99 

0.15 

0.99 

0.16 

0.98 

0.18 

0.98 

0.20 

rasIJtj 

1.23 

0.18 

1.23 

0.21 

1.23 

0.23 

1.22 

0.25 

TABLES. 


59 


TABLE  V '.—Continued. 
HORIZONTAL  DISTANCES  AND  ELEVATIONS  FROM  STADIA  READINGS. 


12° 

13° 

14° 

15° 

Minutes. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

O      .      . 

95.68 

20.34 

94-94 

21.92 

94-15 

23-47 

93-30 

25.00 

2      .      . 

95-65 

20.39 

94.91 

21.97 

94.12 

23-52 

93-27 

25-05 

4     •     • 

95-63 

20.44 

94.89 

22.O2 

94.09 

23-58 

93-24 

25.10 

6     .     . 

95.61 

20.50 

94.86 

22.08 

94-07 

23-63 

93.21 

25.I5 

8     .     . 

95-58 

20-55 

94.84 

22.13 

94.04 

23.68 

93.18 

25.20 

10     .    . 

95-56 

20.60 

94.81 

22.18 

94.01 

23-73 

93.16 

25-25 

12       .       . 

95-53 

2066 

94-79 

22.23 

93-98 

23.78 

93-  i  3 

25-30 

14       .       . 

95-5i 

20.71 

94.76 

22.28 

93-95 

23.83 

93.10 

25-35 

16    .    . 

95-49 

20.76 

94-73 

22.34 

93-93 

23.88 

93-07 

25.40 

18     .     . 

95-46 

20.8  1 

9471 

22.39     93-90 

23-93 

93-04 

25-45 

20      .      . 

95-44 

20.87 

94.68 

22.44 

93-87 

23-99 

93.01 

25-50 

22       ,      . 

95.41 

20.92 

94.66 

22.49 

93-84 

24.04 

92.98 

25-55 

24      .      . 

95-39 

20.97 

94-63 

22.54 

93.81 

24.09 

92-95 

25.60 

26      .      . 

95-36 

21.03 

94.60 

22.60 

93-79 

24.14 

92.92 

25-65 

28      .      . 

95-34 

2  1.  08 

94.58 

22.65 

93-76 

24.19 

92.89 

25.70 

30      .      . 

95-32 

21.13 

94-55 

22.70 

93-73 

24.24 

92.86 

25-75 

32      .      . 

95.29 

2I.I8 

94-52 

22-75 

93-70 

24.29 

92-83 

25.80 

34      •      • 

95-27 

21.24 

94-5° 

22.80 

93-67 

24-34 

92.80 

25.85 

36      •      • 

95-24 

21.29 

94-47 

22.85 

93-65 

24-39 

92.77 

25.90 

38      •      . 

95-22 

21.34 

94-44 

22.91 

93.62 

24.44 

92.74 

25-95 

40      .      . 

95-!9 

21-39 

94.42 

22.96 

93-59 

24.49 

92.71 

26.00 

42      .      . 

95-^7 

21-45 

94-39 

23.01 

93-56 

24-55 

92.68 

26.05 

44     •     • 

95-'4 

21.50 

94-36 

23.06 

93-53 

24.60 

92-65 

26.10 

46     .     . 

95.12 

2i-55 

94-34 

23.11 

93-5° 

24.65 

92.62 

26.15 

48     .     . 

95-09 

21.60 

94-31 

23.16 

93-47 

24.70 

92.59 

26.20 

50     .     . 

95-07 

21.66 

94.28 

23.22 

93-45 

24-75 

92.56 

26.25 

52     .     . 

95-04 

21.71 

94.26 

23.27 

93-42 

24.80 

92-53 

26.30 

54    •     • 

95.02 

21.76 

94.23 

23-32 

93-39 

24-85 

92-49 

26.35 

56    .     . 

94-99 

21.81 

94.20 

23-37 

93-36 

24.90 

92.46 

26.40 

58     .     . 

94-97 

21.87 

94.17 

23.42 

93-33 

24-95 

92-43 

26.45 

60    .     . 

94-94 

21.92 

94-15 

23-47 

93-30 

25.00 

92.40 

26.50 

'  =  0.75 

0-73 

0.16 

0-73 

0.17 

0-73 

0.19 

0.72 

O.2O 

c  =  i.oo 

0.98 

O.22 

0.97 

0-23 

o-97 

0.2S 

0.96 

O.27 

c  =  1.25 

1.22 

0.27 

1.  21 

0.29 

1.  21 

0-31 

i.  20 

0-34 

1 

6o 


SURVEYING. 


TABLE  V.— Continued. 
HORIZONTAL  DISTANCES  AND  ELEVATIONS  FROM  STADIA  READINGS. 


mpf*__^_4._  _. 

16° 

17° 

18° 

19° 

Minutes. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Eiev. 

Dist. 

Elev. 

Dist. 

Elev. 

0     .     . 

92.40 

26.50 

9r-45 

27.96 

9°-45 

29-39 

89.40 

30.78 

*> 

92-37 

26.55 

91.42 

28.01 

90.42 

29.44 

89.36 

30.83 

4    -    • 

92-34 

26.59 

9T-39 

28.06 

90.38 

29.48 

89-33 

30.87 

6    .    . 

92.31 

26.64 

9r35 

28.10 

90.35 

29-53 

89.29 

30.92 

8    .    . 

92.28 

26.69 

91.32 

28.15 

90.31 

29.58 

89.26 

30-97 

10      .      . 

92.25 

26.74 

91.29 

28.20 

90.28 

29.62 

89.22 

31.01 

12      .      . 

92.22 

26.79 

91.26 

28.25 

90.24 

29.67 

89.18 

31.06 

14      .      . 

92.19 

26.84 

91.22 

28.30 

9O.2I 

29.72 

89.15 

31.10 

16    .    . 

92.15 

26.89 

91.19 

28.34 

90.IS 

29.76 

89.11 

3I-I5 

18    .    . 

92.12 

26.94 

91.16 

28.39 

90.14 

29.81 

89.08 

3LI9 

20    .    . 

92.09 

26.99 

91.12 

28.44 

9O.II 

2Q.86 

89.04 

3L24 

22      .      . 

92.06 

27.04 

91.09 

28.49 

90.07 

29.90 

89.00 

31.28 

24      .      . 

92.03 

27.09 

91.06 

28.54 

90.04 

29.95 

88.96 

31-33 

26      .      . 

92.00 

27.13 

91.02 

28.58 

90.00 

30.OO 

88.93 

3I-38 

23      .      . 

91.97 

27.18 

90.99 

28.63 

89.97 

30.04 

88.89 

31-42 

30      .      . 

9T-93 

27.23 

.  90-96 

28.68 

8.9-93 

30.09 

88.86 

3M7 

32  -  • 

91.90 

27.28 

90.92 

28.73 

89.90 

30.14 

88.82 

3i-5i 

34    •    • 

91.87 

27-33 

90.89 

28.77 

89.86 

30.19 

88.78 

3I-56 

36  .  . 

91.84 

27.38 

90.86 

28.82 

89-83 

30.23 

88.75 

31.60 

38  .  . 

91.81 

27-43 

90.82 

28.87 

89.79 

30.28 

88.71 

31-65 

40  .  . 

91.77 

27.48 

90.79 

28.92 

89.76 

30-32 

88.67 

31.69 

42   ,  . 

91.74 

27-52 

90.76 

28.96 

89.72 

30.37 

88.64 

31-74 

44     •     • 

91.71 

27-57 

90.72 

29.01 

89.69 

30.41 

88.60 

3I-78 

46    .     . 

91.68 

27.62 

90.69 

29.06 

89.65 

30.46 

88.56 

31-83 

48     .    . 

91.65 

27.67 

90.66 

29.11 

89.61 

30-  5  i 

88.53 

31-87 

50    .     . 

91.61 

27.72 

90.62 

=9-15 

89.58 

30-55 

88.49 

31.92 

52     .     . 

91.58 

27.77 

90-59 

29.20 

89.54 

30.60 

88.45 

31.96 

54    .     - 

9^-55 

27.81 

90-55 

29.25 

89.51 

30-65 

88.41 

32.01 

56    .     . 

91-52 

27.86 

90.52 

29.30 

89.47 

30-69 

88.38 

3--05 

58    .  $ 

91.48 

27.91 

90.48 

29-34 

89.44 

30-74 

88.34 

32.09 

60    .  :.;' 

9M5 

27.96 

90-45 

29-39 

89.40 

30.78 

88.30 

32*14 

r  =  o.75 

0.72 

O.2I 

0.72 

0.23 

0.71 

0.24 

0.71 

0.25 

f  =  i  .00 

0.86 

0.28 

o-95 

0.30 

°-95 

0.32 

0-94 

Q-33  [ 

'  =  1.25 

i.  20 

0-35 

1.19 

0.38 

1.19 

0.40 

1.18 

0.42  J 

TABLES. 


6l 


TABLE  V .—Continued. 
HORIZONTAL  DISTANCES  AND  ELEVATIONS  FROM  STADIA  READINGS. 


2< 

>° 

2: 

1° 

2 

2° 

2J 

*° 

Minutes. 

Hor. 

DifT. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

o    .     . 

88.30 

32.14 

87.16 

33-46 

85.97 

34-73 

84-73 

35-97 

2      ,      . 

88.26 

32.18 

87.12 

33-50 

85-93 

34-77 

84.69 

36.01 

4     •     • 

88.23 

32.23 

87.08 

33-54 

85.89 

^4.82 

84.65 

36-05 

6    .     . 

88.19 

32.27 

87.04 

33-59 

85.85 

34.86 

84.61 

36.09 

8    .     . 

88.15 

32-32 

87.00 

33-63 

85.80 

34.90 

84.57 

36-13 

10    .     . 

88.11 

32.36 

86.96 

33-6? 

85-76 

34-94 

84.52 

36-17 

12      .      . 

88.08 

32.4I 

86.92 

33-72 

85.72 

34.98 

84.48 

36.21 

14      .      „' 

88.04 

32.45 

86.88 

33-76 

85.68 

35.02 

84.44 

36-25 

16    .    . 

88.00 

3249 

86.84 

33-8o 

85-64 

35-07 

84.40 

36.2Q  | 

18    .    . 

87.96 

32.54 

86.80 

33-84 

85.60 

35-11 

84-35 

36-33 

20      .      . 

87-93 

32.58 

86.77 

33-89 

85.56 

35-  i  5 

84.31 

36.37 

22      .      . 

87.89 

32.63 

86.73 

33-93 

85.52 

35-19 

84-27 

36.41 

24      .      . 

87.85 

32.67 

86.69 

33-97 

85.48 

35-23 

84-23 

36.45 

26      .      . 

87.81 

32.72 

86.65 

34.01 

85-44 

35-27 

84.18 

36.49 

28      .      . 

87.77 

32.76 

86.61 

34.o6 

8540 

35-31 

84-14 

36.53 

30      .      . 

87.74 

32.80 

86.57 

34.10 

85.36 

35-36 

84.10 

36.57 

32      .      . 

87.70 

32.85 

86.53 

34-M 

85-3I 

35-40 

84.06 

36.61 

34    •    . 

87.66 

32.89 

86.49 

34.18 

85.27 

35-44 

84.01 

36.65 

36    •    . 

87.62 

32.93 

8645 

34.23 

85-23 

3548 

83-97' 

36.69 

38    •    • 

87.58 

32.98 

86.41 

34.27 

85.19 

35-S2 

83-93 

36.73 

40    .    . 

87.54 

33-02 

86.37 

34-3« 

85.15 

35-56 

83-89 

36.77 

42    .    . 

87-51 

33-07 

86.33 

34-35 

85.11 

35.60 

83-84 

36.80 

44     •     • 

87.47 

33-n 

86.29 

34-40 

85.07 

35-64 

83.80 

36.84 

46     .     . 

8743 

33-  J  5 

86.25 

34-44 

85.02 

35-68 

83-76 

36.88 

48     .     . 

87.39 

33-20 

86.21 

34-48 

84-98 

35=72 

83.72 

36.92 

50     .     . 

87-35 

33-24 

86.17 

34-52 

84.94 

35-76 

83.67 

36.96 

52     .     . 

87.31 

33.28 

86.13 

34-57 

84.90 

35.80 

83.63 

37-oo 

54    -     • 

87.27 

33-33 

86.09 

34.61 

84.86 

35-85 

83.59 

37-04 

56    .    . 

87.24 

33-37 

86.05 

34.65 

84.82 

35.89 

83-54 

37.08 

58     .    . 

87.20 

33-41 

86.01 

34.69 

84.77 

35-93 

83.50 

37.12 

60    .     . 

87.16 

3346 

85.97 

3473 

84.73 

35-97 

83.46 

37-i6 

<r  =  0.75 

0.70 

0.26 

0.70 

0.27 

0.69 

0.29 

0.69 

0.30 

C  =  I.OO 

0.94 

o-35 

o-93 

0-37 

0.92 

0.38 

0.92 

0.40 

rai.25 

1.17 

0.44 

1.16 

0.46 

I.IS 

0.48 

MS 

0.50 

62 


SUR  VE  YING. 


TABLE  V.— Continued. 
HORIZONTAL  DISTANCES  AND  ELEVATIONS  FROM  STADIA  READINGS. 


r—- 

24° 

25° 

26° 

27° 

Minutes. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist 

Elev. 

Dist. 

Elev. 

Dist. 

Elev 

Dist. 

Elev. 

o    .     . 

83.46 

37.16 

82.14 

38.30 

80.78 

39-40 

79-39 

40.45 

2      .      . 

83.41 

37.20 

82.09 

38.34 

80.74 

39-44 

79-34 

40.49 

4     •     • 

83-37 

37-23 

82.05 

38.38 

80.69 

39-47 

79-30 

40.52 

6    .    . 

83-33 

37-27 

82.01 

38.41 

80.65 

39-51 

79-25 

40.55 

8    .    . 

83.28 

37-31 

81.96 

38.45 

80.60 

39-54 

79.20 

40-59 

10      .       . 

83.24 

37-35 

81.92 

38.49 

80-55 

39-58 

79-  i  5 

40.62 

12      .      . 

83.20 

37-39 

81.87 

38.53 

80.51 

39.61 

79.11 

40.66 

14      .      . 

83-I5 

37-43 

81.83 

38.56 

80.46 

39.65 

79.06 

40.69 

16    .     . 

83.11 

37-47 

81.78 

38.60 

80.41 

39-69 

79.01 

40.72 

18    .    . 

83.07 

37-51 

81.74 

38.64 

80.37 

39-72 

78.96 

40.76 

20    .    . 

83.02 

37-54 

81.69 

38.67 

80.32 

39-76 

78.92 

40.79 

22      .      . 

82.98 

37-5S 

81.65 

38.71 

80.28 

39-79 

78.87 

40.82 

24      .      . 

82.93 

37-62 

81.66 

3875 

80.23 

39-83 

78.82 

40.86 

26      .      . 

82.89 

37.66 

81-56 

38.78 

80.  1  8 

39-86 

78.77 

40.89 

28      .      . 

82.85 

37-70 

81.51 

38.62 

80.14 

39-90 

78.73 

40.92 

30      .      . 

82.80 

37-74 

81.47 

38.86 

80.09 

39-93 

78.68 

40.96 

32      .      . 

82.76 

37-77 

81.42 

38.89 

80.04 

39-97 

78.63 

40.99 

34    •    • 

82.72 

37-81 

81.38 

38.93 

80.00 

40.00 

78.58 

41.02 

36    .    . 

82.67 

37.85 

81-33 

38.97 

79-95 

40.04 

78.54 

41.06 

33    -    . 

82.63 

37.89 

81.28 

39.00 

79.90 

40.07 

78.49 

41.09 

40    .    . 

82.58 

37-93 

81.24 

39-°4 

79.86 

40.11 

78.44 

41.12 

42    .    . 

82.54 

37.96 

81.19 

39.08 

79.81 

40.14 

78.39 

41.16 

44    •    - 

82.49 

38.00 

81.15 

39-  Ir 

79.76 

40.18 

78.34 

41.19 

46    .     . 

82.45 

38.04 

Sr.io 

39-  i  5 

79.72 

40.21 

78.30 

41.22 

48    .    . 

82.41 

38.08 

8  1.  06 

39.18 

79-67 

40.24 

78-25 

41.26 

50    .     . 

82.36 

38.11 

81.01 

39.22 

79.62 

40.28 

78.20 

41.29 

.   52    •    • 

82.32 

38-15 

80.97 

39.26 

79-58 

40.31 

78.15 

41.32 

54    .    • 

82.27 

38.19 

80.92 

39.29 

79-53 

40-35 

78.10 

41-35 

56    .    . 

82.23 

38.23 

80.87 

39-33 

79.48 

40.38 

78.06 

4L39 

58    .    . 

82.18 

38.26 

80.83 

39-36 

79-44 

40.42 

78.01 

41.42 

60    .    . 

82.14 

38.30 

80.78 

39-40 

79-39 

40.45 

77.96 

41.45 

'  =  0.75 

0.68 

0.31 

0.68 

0.32 

0.67 

°-33 

0.66 

°-35 

<:=  1.00 

0.91 

0.41 

0.90 

o-43 

0.89 

o-45 

0.89 

0.46 

'=1.25 

1.14 

0.52 

*««3 

0-54 

1.  12 

0.56 

i.  ii 

0.58  ;' 

TABLES. 


TABLE  V '.—Continued. 
HORIZONTAL  DISTANCES  AND  ELEVATIONS  FROM  STADIA  READINGS. 


28° 

29° 

3O° 

Minutes. 

Hor. 

Diff, 

Hor. 

Diff. 

Hor. 

Diff. 

Dist: 

Elev. 

Dist 

Elev. 

Dist. 

Elev. 

v 

o     .     . 

77,96 

41-45 

76.50 

42.40 

75-0° 

43.30 

2      .      . 

77-91 

41.48 

76.45 

4243 

74.95 

43-33 

4     •     • 

77.86 

41.52 

76.40 

42.46 

74.90 

43.36 

6    .     . 

77.81 

41.55 

76.35 

42.49 

74.85 

43-39 

8    .     . 

77-77 

41.58 

76.30 

42-53 

74.80 

4342 

10       .      . 

77.72 

41.61 

76.25 

42.56 

74-75 

43-45 

12       .       . 

77.67 

41.65 

76.20 

42.59 

74.70 

4347 

14       .       . 

77.62     . 

41.68 

76.15 

42.62 

74.65 

43-50 

16    .    . 

77-57 

41.71 

76.10 

42.65 

74.60 

43-53 

18    .    . 

77-S2 

41.74 

76.05 

42.68 

74-55 

43-56 

20      .      . 

77.48 

41-77 

76.00 

42.71 

74-49 

43-59 

22       .      . 

77.42 

4I.8l 

75-95 

42.74 

74-44 

43.62 

24      .      . 

77.38 

41.84 

75-90 

42.77 

74-39 

43-65 

26      .      . 

77-33 

41.87 

75-85 

42.80 

74-34 

43-67 

28      .      . 

77.28 

41.90 

75.80 

42.83 

74.29 

43-70 

3° 

77.23 

4L93 

75-75 

42.86 

74.24 

43-73 

32 

77.18 

41.97 

75-70 

42.89 

74.19 

43-76 

34     •     • 

77.I3 

42.OO 

75-65 

42.92 

74.14 

43-79 

36    .    . 

77.09 

42.03 

75.60 

42.95 

74.09 

43.82 

38     .    . 

77.04 

42.06 

75-55 

42.98 

74.04 

43-84 

40     .     . 

76.99 

42.09 

75-5° 

43-01 

73-99 

43-87 

42     .     . 

76.94 

42.12 

75-45 

43-°4 

73-93 

43-9° 

44     •     • 

76.89 

42.15 

75-40 

43-°7 

73.88 

43-93 

46     .     . 

76.84 

42.19 

75-35 

43.10 

73-83 

43-95 

48     .     . 

76.79 

42.22 

75.30 

43-  r  3 

73-78 

43-98 

50     .     . 

76.74 

42.25 

75-25 

43.16 

73-73 

44.01 

52     .     . 

76.69 

42.28 

75.20 

43.18 

73-68 

44.04 

54     .    . 

76.64 

42.3I 

7S-*S 

43-21 

73-63 

44-07 

56    .    . 

76.59 

42-34 

75-10 

43-24 

73-58 

44.09 

58     .    . 

76.55 

42.37 

75.05 

43-27 

73-52 

44.12 

60    .    . 

76.50 

42.40 

75.00 

43-30 

73-47 

44.I5 

fa  075 

0.66 

0.36 

0.65 

o-37 

0.65 

0.38 

c  —  i.oo 

0.88 

0.48 

0.87 

0.49 

0.86 

0.51 

<r  =  1.25 

1.  10 

0.00 

1.09 

0.62 

1 

1.08 

0.64 

64 


SURVEYING. 


TABLE  VI. 

NATURAL  SINES  AND  COSINES. 


Sine 

.01745 
.01774 
.01803 
.01832 
.01862 
.01891 
.01920 
.01949 
.01978 
.02007 


.00262  One. 
.00291  One. 


02065 
02094 
02123 
02152 
02181 
02211 
02240 
02269 
02298 
02327 


99979 
99978 
99977 
99977 
99976 
99976 
99975 
99974 
99974 


07295 
07324 
07353 
07382 
07411 
07440 


00849 

00378 
00407 
00436 
00465 
00495 
00524 
00553 
00582 


07585 
07614 
(17643 
07672 
07701 
07730 
07759 
07788 
07817 
07846 


04100 
0-1129 
04159 
04188 
04217 
04246 
04275 
04304 
04333 


02385 
02414 
02443 
02472 
02501 
02530 
02560 
02589 
02618 


99827 
99826 
99824 
99622 


05960 

05989 
OG018 
OG047 
06076 
06105 


99819 
99817 
99815 
99813 


02647 
02676 
02705 
02734 
02763 
02792 
02821 
02850 
02879 
02908 


04391 
04420 
04449 
04478 
04507 
04536 
04565 
04594 
04623 
04653 


06134 

06163 
06192 
06221 
06250 
06279 
06308 
06337 


07875 
07904 
07933 
07962 
07991 


99963 

99962 
99961 
99960 
99959 
99959 
99958 

99957 
99956 
99955 
99954 
99953 


01018 
01047 
01076 
01105 
01134 
01164 

01193 
01222 
01251 

01280 


08049 
08078 
08107 
08136 


04682 
04711 
04740 
04769 
04798 
04827 
04856 
04885 
04914 
04943 


08165 
08194 
08223 
08252 
08281 
08310 
08339 
08368 
08397 
08426 


06453 
06482 
06511 
06540 
06569 
06598 
06627 
06656 


02967 
02996 
03025 
03054 
03083 
03112 
03141 
03170 
03199 


.01338 
.01367 
.01396 
.01425 
.01454 


.99784 
.99782 
.99780 
.99778 
.99776 


.08455  .99642 
.08484  .99639 


06714 
06743 
06773 
06802 
06831 


.99943 
.99942 
.99941 
.99940 


.03403 
.03432 
.03461 
.03490 
Cosin 


.01658 

.99986 
.017161.99985 
.017451.99985 


89° 


TABLES. 


TABLE  VI.—  Continued. 
NATURAL  SINES  AND  COSINES. 


40 


Sine 


.08745 
.08774 


Cosin 
799619 
.99617 


.08831 


.08918 
.08947 
.08976 


.09034 
.09063 
.09092 
.09121 
.09150 
.09179 


.09237 
.09266 
.09295 

.09324 
.09353 


.09411 
.09440 


.09527 
.09556 

.09585 

.09614 

.09642 
.09671 
.09700 
.09729 
.09758 
.09787 
.09816 
.09845 


.09932 
.09961 
.09990 
.10019 
.10048 
.10077 
.10106 
.10135 
.10164 

.10192 

.10221 
.10250 


.99612 


.996071 
.99604 | 
99602 
.99599! 
.99596 


99591 
99588 
99586 
99583 
99580 
99578 
99575 
99572 
99570 
99567 

99564 
99562 
99559 
99556 
99553 
99551 
99548 
99545 
99542 
99540 

,99537 
,99534 
.99531 


99526 


99520 
99517 
99514 
99511 


.99506 
.99503 
.99500 
.99497 
.99494 
.99491 
.99488 
.99485 
.99482 

.99479 
.99476 


.10279  .G9470 


.99464 


.10337 
10366 
.10395 
.10424 
.10453 


Cosin 


.99458 
.99455 


Sine 


84° 


Sine_ 
.10453 
.10482 
.10511 
.10540 


.10597 
.10626 
.10655 
,10684 
,10713 
,10742 

,10771 
,10800 
.10829 
.10858 
.10887 
.10916 
.10945 
.10973 
.11002 
.11031 

.11060 
.11089 
.11118 
.11147 
.11176 
.11205 
.11234 
.11263 
.11291 
.11320 

.11349 
.11378 
.11407 
.11436 
.11465 
.11494 
.11523 
.11552 
.11580 
.11609 


.11667 
.11696 
.11725 
.11754 
.11783 
.11812 
.11840 
.11869 
.11898 

.11927 
.11956 
.11985 
.12014 
.12043 
.12071 
I .12100 
.12129 
, .12158 
I .12187 
Cosin 


Cosin 


.99452 
.99449 
.99446 
.99443! 


.99437 
.99434 
.99431 
.99428 
.99424 
.99421 

.99418 
.99415 
.99412 
.99409 
.99406 
.99402 


,99396 
,99393 


99377 
99374 
99370 
99367 
99364 
993SO 
99357 

99354 
99351 
99347 
90344 
99341 
99337; 
99334; 
99331 j 
99327 
99324, 


99317, 
,99314 
,99310 
99307 
,99303 
99300 
,99297 


.99279 
.99276 
.99272 


.99265 


Sine 


83< 


Sine 


.12187 
.12216 
.12245 
.12274 
.12302 
.12331 
.12360 


Cosin 


.99255 
.99251 


.99244 
.99240 


.12418 
.12447 
.12476 

.12504 
.12533, 
.12562 
.12591 
.12620 
.12649 
.12678 
.12706 
.12735 
.12764 

.12793 

.12822 
.12351 
.12380 
.12908 
.12937 
.12966 
.12995 
.13024 
.13053 

.13081 
.13110 
.13139 
.13168 
.13197 
.13226 
.13254 
.13283 
.13312 
.13341 

.13370 


.13427 
.13456 
.13485 
.13514 
.13543 
.13572 
.13600 


.13658 
.13687 
.13716 
.13744 
.13773 
.13802 
.13831 
.13860 


.13917 


Cosin 


.99230 
.99226 
.99222 
.99219 

.99215 
.99211 

99208 | 
,992041 
99200 
,99197 
,99193 
.99189 
,99186 


.99178 
.99175 
.99171 
.99167 
.991C3 
.99100 
.99156 
.99152 
.99148 
.99144 

,99141 
,99137 
.99133 
.99129 
.90125 
.99122 
.99118 
.99114 
.99110 
.99106 

.99102 


.99094 
.99091 
.99087 
.99083 
.99079 
.99075 
.99071 
.99067 


.99059 
.99055 
.99051 
.99047 
.99043 


.99035 
.99031 
^99027 
Sine 


II 


82° 


Cosin 


.99027 
.99023 
.99019 
.99015 
.99011 


.98990 


.98965 


.98953 
.98948 
.98944 


.98927 
.98923 
.98919 


Sine 

.13917 
.13946 
.13975 
.14004 
.14033 
.14061 
.14090 
.14119 
.14148 
.14177 
.14205 

.14234 
.14263 
.14292 
.14320 
.14349 
.14378 
.14407 
.14436 
.14464 
.14493 

.14522 
.14551 
.14580 
.14608 
.14637 
.14666 
.14695 
.14723 
.14752 
.14781 


.14810 

.14838 

.  14867  \.vww 

.148961.98884 

.14925 

.14954 

.14982 

.15011 

.15040 

.15069 

.15097 
.15126 
.15155 
.15184 
.15212 
.15241 
.15270 
.15299 
.15327 
.15356 

.15385 
.15414 
.15442 
.15471 
.15500 
.15529 
.15557 
.15586 
.15615 
.15643 


.98906 
.98902 


Cosin 


.98880 
.98876 
.98871 
.98867 
.98863 


.98849 
.98845 
.98841 


.98827 


.98796 
.98791 
.98787 
.98782 
.98778 
.98773 


Sine 


81s 


.98769 
.98764!  59 
.98760  58 
.98755  57 


jsine  I  Cosin 

715643 

.15672 

.15701 

.15730 

.15758 

.15787 

.15816 

.15845 

.15873 

.15902 

.15931 

.15959 
.15988 
.16017 
.16046 
.16074 
.16103 
.16132 
.16160 
.16189 
.16218 


.16275 


.16361 


,16419 
,16447 
,16476 
,16505 

,16533 
,16562 
,16591 
,16620 
,16648 
,16677 
,16706 
,16734 
,16763 
.16792 

.16820 
.16849 
.16878 
.16906 
.16935 
.16964 


.17021 
.17050 
.17078 

.17107 
.17136 
.17164 
.17193 
.17222 
.17250 
.17279 
.17308 
.17336 
.17365 


Cosin 


.98751 
.98746 
.98741 
.98737 
.98732 
.98728 
.98723 

.98718 
.98714 
.98709 
.98704 


.98700!  45 
.98695 !  44 
.98690;  43 
.98686;  42 
.98681!  41 
. 98676 j  40 

.98671 !  39 

.  98667  i  38 
.98662,  37 
.98657  36 
.98652  35 
.98648!  34 
.98643  33 
.98638  32 
.98633  31 


,98600 
.98595 
.98590 

.08585 


98624  29 
98619;  23 
98614!  27 
98609',  26 
25 
24 
23 
22 
21 
20 

19 
28 
17 
16 
15 
14 
13 
12 


.98575 
.98570 
.98565 
.98561 
.98556 
.98551 


,98541 
,98536 
,98531 


,98521 
,98516 
,98511 


,98501 
,98496 
,98491 


98481 


Sine 


66 


SURVEYING. 


TABLE  VI.  —Continued. 
NATURAL  SINES  AND  COSINES. 


10° 

11° 

12° 

13° 

14° 

Sine  Cosin 

Sine 

Cosin 

Sine  i  Cosin 

Sine  |  Cosin 

Sine  1  Cosin 

' 

0 

.17365  .98481 

.19081 

.98163 

.20791  .97815 

.22495 

.97437 

.24192 

.97030 

60 

1 

.17393  .98476 

.19109 

.98157 

.20820 

.97809 

.22523 

.97430 

.24220 

.97023 

59 

2 

.17422  .98471 

.19138 

.98152 

.20848 

.97803 

.22552 

.97424 

.24249 

.97015 

5R 

3 

.17451  .98466 

.19167 

.98146 

.20877 

.97797 

.22580 

.97417 

.24277 

.97008!  fi7 

4  i.  17479  ;.  98461 

.19195 

.98140 

.20905 

.97791 

.22608 

.97411 

.24305 

.97001 

56 

5  i.  17508  .98455 

.19224 

.98135 

.20933 

.97784 

.22637 

.97404 

.24333 

.96994 

55 

61.17537  .98450 

.19252 

.98129 

.20962 

.97778 

.22665 

-G7398 

.24362 

.96987 

54 

7 

.17565  .98445 

.19281 

.98124 

.20990 

.97772 

.22693 

.97391 

.24390 

.96980 

53 

8 

.17591  .98440 

.19309 

.98118! 

.21019 

.97766 

.22722 

.97384 

.24418 

.96973 

52 

9 

.17623  .98435 

.19338 

.98112! 

.21047 

.97760 

.22750 

.97378 

.24446 

.96966 

51 

10 

.17651  .98430 

.19366 

.98107! 

.21076 

.97754 

.22778 

.97371 

.24474 

.96959 

50 

11 

.17680 

.98425 

.19395 

.98101 

.21104 

.  97748  ! 

.22807 

.97365 

.24503 

.96952 

49 

.17708 

.98420 

.19423 

.98096 

.21132 

.97742 

.22835 

.97358 

.24531 

.96945 

48 

13 

.17737 

.98414 

.19452 

.98C90! 

.21161 

.97735' 

.22863 

.97351 

.24559 

.96937 

47 

14 

.17766 

.98109 

.19481 

.98084 

.21189 

.97729 

.22892 

.97345 

.24587 

.96930 

46 

15 

.17794 

.98404 

.19509 

.98079; 

.21218 

.97723 

.22920 

.97338 

.24615 

.96923 

45 

16 

.17823 

.98399 

.19538 

.98073 

.21246 

.97717 

.22948 

.97331 

.24644 

.96916 

44 

17 

.17852 

.98394 

.19566 

.98067 

.21275 

.97711 

.22977 

.97325 

.24672 

.96909 

43 

18 

.17880 

.98389 

.19595 

.98061 

.21303 

.97705 

.23005 

.97318 

.24700 

.96902 

42 

19 

.17909 

.98383 

.19623 

.98056 

.21331 

.97698 

.23033 

.97311 

.24728 

.96894 

41 

20 

.17937 

.98378 

.19652 

.98050! 

.21360 

.97692 

.23062 

.97304 

.24756 

.96887 

40 

21 

.17966 

.98373 

.19680 

.98044 

.21388 

.  97686  ' 

.23090 

.97298 

.24784 

.96880 

39 

22 

.17995 

.98368 

.19709 

.98039 

.21417 

.97680 

.23118 

.97291 

.24813 

.96873 

38 

23 

.18023 

.98362 

.19737 

.93033! 

.21445 

.97673 

.23146 

.97284 

.24841 

.96866 

37 

24 

.18052 

.98357 

.19766 

.98027 

.21474 

.97667 

.23175 

.97278 

.24869 

.96858 

36 

25 

.18081 

.98352 

.19794 

.98021 

.21502 

.97661! 

.23203 

.97271 

.24897 

.96851 

35 

26 

.18109 

.98347 

.19823 

.98316' 

.21530 

.97655 

.23231 

.97264 

.24925 

.96844 

34 

27 

.18138 

.98341 

.19851 

.93310 

.21559 

.97648 

.23260 

.97257 

.24954 

.96837 

33 

28 

.18166 

.98336 

.19880 

.  93004  : 

.21587 

.97642' 

.23288 

.97251 

.24982 

.96829 

32 

29 

.18195 

.93331 

.19908 

.97933 

.21616 

.97633 

.23316 

.97244 

.25010 

.96822;  31 

30 

.18224 

.98325 

.19937 

.97992 

.21644 

.97630, 

.23345 

.97237 

.25038 

.96815 

30 

31 

.18252 

.98320 

.19965 

.97937' 

.21672 

97623  ! 

.23373 

.97230 

.25066 

.96807 

29 

32 

.18281 

.98315 

.19994 

97981 

.21701 

.97617! 

.23401 

.97223 

.25094 

.96800 

23 

33 

.18309 

.98310 

.20022 

.97975 

.21723 

.97611 

.23429 

.97217 

.25122 

.96793 

27 

34 

.18333 

.93304 

.20051 

.97963 

.21758 

.97834 

.23458 

.97210 

.25151 

.96786 

23 

35 

.18367 

.98299 

.20079 

97963 

.21788 

97598 

.23486 

.97203 

.25179 

.96778 

25 

36 

.18395 

.93294 

.23103 

97953 

.21814 

97592 

.23514 

.97196 

.25207 

.96771 

24 

37 

38 

.18424 
.18452 

Si 

.20136 
.20165 

97952 
97946 

.21843 

.21871 

97535 
.97579 

.23542 
.23571 

.97189 
.97182 

.25235 
.25263 

.96764 
.96756 

23 
22 

39  .18481 

.93277 

.20193 

97940: 

.21899 

97573 

.23599 

.97176 

.25291 

.96749 

21 

40 

.18509 

.98272 

.20222 

97934 

.21928 

97566 

.23627 

.97169 

.25320 

.96742 

20 

41 

.18538 

.98267 

.20250 

97928 

.21956 

.97560 

.23656 

.97162 

.25348 

.96734 

19 

42 

.18567 

.93261  I  .20279 

97322; 

.219S5 

.97553 

.23684 

.97155 

.25376 

.96727 

IS 

43 

.18595 

.93256  .20307 

97916 

.22013 

.97547 

.23712 

.97148 

.25404 

.96719 

.18624 

.98250  .20336 

97310 

.22041 

.97541 

.23740 

.97141 

.25432 

.96712 

16 

45 

.18652 

.98245 

.20364 

97305  i 

.22070 

.97534 

.23769 

.97134 

.25460 

.96705  15 

48 

.18681 

.93240 

.20393 

97899 

.22093 

.97528 

.23797 

.97127ii  .25488  .96697  14 

47 

.18710 

.93231!  .20421 

.97893 

.22126 

.07521 

.23825 

.971201!  .25516  .  96690  j  13 

43 

.18738 

.93229 

.20450 

.97887 

.22155 

.97515] 

.23853 

.97113  .25545 

.96682;  12 

49  .18767 

.98223 

.20478 

.97881 

.22183 

.97508 

.23882 

.97106  .25573 

.96675!  11 

50  .18795 

.98218  .20507 

.97875 

.22212 

.97502 

.23910 

.97100  i  .25601 

.96667 

10 

51  .18824 

.98212  .20535 

.97869 

.22240 

.97496 

.23938 

.97093 

.25629  .96660 

9 

52  .18852 

.98207;  .20563 

.97863 

.22268 

.97489! 

.23966 

.97086 

.25657 

.96653 

8 

53  .18881 

.98201 

.20592 

.97857 

.22297 

.97483] 

.23995 

.97079 

.25685 

.96645 

7 

54 

.18910 

.98196 

.20620 

.97851 

.22325 

.  97476  j 

.24023 

.97072 

.25713 

.96638 

6 

55 

.18938 

.98190 

.20649 

.97845 

.22353 

.97470: 

.24051 

.970651 

.25741 

.96630 

5 

56 

.18967 

.98185 

.20677 

.97839 

.22382 

.97463 

.24079 

.97058 

.25769  .96623 

4 

57 

.18995 

.98179 

.20706 

.97833 

.22410 

.97457 

.24108 

.97051 

.25798  .96615  3 

58 

.19024  .98174 

.20734 

.97827 

.22438 

.97450 

.24136 

.97044 

.25826  .  96608  j  2 

59 

.19052  .98168 

.20763 

.97821  .22467 

.97444 

.24164 

.97037 

.25854  .96600 

1 

60 

.19081|.  9816311.  20791 

.97815 

.22495 

.97437 

.24192 

.97030 

.25882  .96593 

0 

Cosin  |  Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

/ 

79° 

78°    '    77°       76°       75° 

TABLES. 


67 


TABLE  VI.  —  Continued. 
NATURAL  SINES  AND  COSINES. 


15° 

16° 

17° 

18° 

19° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

o 

.25882 

.96593 

.27564 

.96126 

.29237 

.95630  .30902 

.95106 

.32557 

.94552 

60 

1 

.25910 

.96585 

.27592 

.96118 

.29265 

.95622  .30929 

.95097 

.32584 

.945421  59 

2 

.25938 

.96578 

.27620  .96110 

.29293 

.95613!  1.30957 

.95088 

.32612 

.94533  58 

3 

.25966 

.96570 

.27648  .96102 

.29321 

.95605  .30985 

.95079 

.32639 

.94523  57 

4 

.25994 

.96562 

.27676  .96094 

.29348 

.  95596  !!.  31012 

.95070 

.32667 

.94514 

56 

5 

.26022 

.96555 

.27704  .96086 

.29376 

.  95588  ii.  31040 

.95061 

.32694 

.94504 

55 

6 

.26050 

.96547 

.27731  .96078  .29404 

.95579 

.31068 

.95052 

.32722 

.94495 

54 

7 

.2G079 

.96540 

.27759:  -96070J  .29432 

.95571 

.31095 

.95043 

.32749 

.94485 

53 

8 

.26107 

.96532 

.27787  .96062!  1.29460 

.95562 

.31123 

.95033 

.32777 

.94476 

52 

9 

.26135 

.96524 

.27815  '.96054 

.29487 

.95554 

.31151 

.95024 

.32004 

.94466 

51 

10 

.26163 

.96517 

.27843 

.96046 

.29515 

.95545 

.31178 

.95015 

.32832 

.94457 

50 

11 

.26191 

.96509 

.27871 

.96037 

.29543 

.95536 

.31206 

.95006 

.328591.94447 

49 

12 

.26219 

.96502 

.27899 

.96029 

.29571 

.95528 

.31233 

.94997 

.32887  .94438 

48 

13 

.26247 

.96194 

.27927 

.96021 

.29599 

.95519 

.31261 

.94988 

.32914  .94428 

47 

14 

.26275 

.96483 

.27955 

.96013 

.29626 

.95511 

.31289 

.94979 

.32942  .94418 

46 

15 

.26303 

.96479 

.27983 

.96005 

.29654 

.95502 

.31316i.  94970 

.32969 

.94409 

45 

16 

.26331 

.96471 

.28011 

.95997 

.29682 

.95493 

.313441.94961 

.32997 

.94399 

44 

17 

.26359 

.96463 

.28039 

.95989 

.29710 

.95485 

.31372 

.94952 

.33024 

.94390 

43 

18 

.26387 

.96456 

.28067 

.95981 

.29737 

.95476 

.31399 

.94943 

.33051 

.94380 

42 

19 

.26415 

.96448 

.28095 

.95972 

.29765 

.95467 

.31427 

.94933 

.33079 

.94370 

41 

20 

.26443 

.96440 

.28123 

.95964 

.29793 

.95459 

.31454 

.94924 

.33106 

.94361 

40 

21 

.26471 

.96433 

.28150 

.95956 

.29821 

.95450 

.31482 

.94915 

!.33134 

.94351 

39 

go 

.26500 

.96425 

.28178 

.95948 

.29849 

.95441 

.31510 

.94906 

i  .33161 

.94342 

38 

23 

.26*28 

.96417 

.28206 

.95940 

.29876 

.95433 

.31537 

.94897 

.33189 

.94332 

37 

24 

.26556 

.96410 

.28234 

.95931 

.29904 

.95424 

.31565 

.94888 

.33216 

.94322 

36 

25 

.26584 

.96402 

.28262 

.95923 

.29932 

.95415 

.31593 

.94878 

.33244 

.94313 

35 

26 

.26612 

.96394 

.28290 

.95915 

.29960 

.95407 

.31620 

.94869 

.33271 

.94303 

34 

27 

.26640 

.96386 

.283181.95907 

.29987 

.95398 

.31648 

.94860 

.33298 

.94293 

33 

28 

.26668 

.96379 

.28340 

.95898 

.30015 

.95389 

.31675 

.94851 

.33326 

.94284 

32 

29 

.26696 

.96371 

.28374 

.95890 

.30043 

.95380 

.31703 

.94842 

.33353 

.94274 

31 

30 

.26724 

.96383 

.28402 

.95882 

.30071 

.95372 

.31730 

.94S32 

.33381 

.94264 

30 

81 

.26752 

.96355 

.28429 

.95874 

.30098 

.95363 

.31758 

.94823 

.33408 

.94254 

29 

32 

.26780 

.96347 

.28457 

.95805 

.30120 

.95354 

.31786 

.94814 

.33436 

.94245 

28 

33 

.26808 

.96340 

.28485 

.95857 

.30154 

.95345 

.31813 

.94805 

.33463 

.94235 

27 

34 

.26836 

.96332 

.28513 

.95849 

.30182 

.95337 

.31841 

.94795 

.33490 

.94225 

26 

35 

.26864 

.96324 

.28541 

.95G41 

.30209 

.95328 

.31868 

.94786 

.33518 

.94215 

25 

36 

.26892 

.96310 

.28569 

.95832 

.30237 

.95319 

.31896 

.94777 

.33545 

.94206 

24 

37 

.26920 

.96303 

.28597 

.95824 

.30265 

.95310 

.31923 

.94768 

.33573 

.94196 

23 

38 

.26948 

.96301 

.28625 

.95816 

.30292 

.95301 

.31951 

.94758 

.33600 

.94186 

22 

39 

.26976 

.96203 

.28652 

.95807 

.30320 

.95293 

.31979 

.94749 

.33627 

.94176 

21 

40 

.27004 

.96285 

.28680 

.95799 

.30348 

.95284 

.32006 

.94740 

.33655 

.94167 

20 

41 

.27032 

.9627? 

.28708 

.95791 

.30376 

.95275 

.32034 

.94730 

.33682 

.94157 

19 

42 

.27060 

.96269 

.28736 

.95782 

.30403 

.95266 

.32061 

.94721 

.33710 

.94147 

18 

43 

.27088 

.96261 

.28764 

.95774 

.30431 

.95257 

.32089 

.94712 

]  .38737 

.94137 

17 

44 

.27116 

.96253 

.28792 

.95766 

.30459 

.95248 

.32116 

.94702 

.33764 

.94127 

16 

45 

.27144 

.96246 

.28820 

.95757 

.30486 

.95240 

.32144 

.94693 

.33792 

.94118 

15 

46 

.27172 

.96238 

.28847 

.95749 

.30514 

.95231 

.32171 

.94684 

1.33819 

.94108 

14 

47 

.27200 

.96230 

.28875 

.95740 

.30542 

.95222 

.32199 

.94674 

.33846 

.94098 

13 

48 

.27228 

.96222 

.28903 

.95732 

.30570 

.95213 

.32227 

.94665 

.33874 

.94088 

12 

49 

.27256 

.96214 

.28931 

.95724 

.30597 

.95204 

.32254 

.94656 

33901 

.94078 

11 

50 

.27284 

.96206 

.28959 

.95715 

i.  30625 

.95195 

.32282 

.94646 

.33929 

.94068 

10 

51 

.27312 

.96198 

.28987 

.95707 

1  .30653 

.95186 

.32309 

.94637 

.33956 

.94058 

9 

52 

.27340 

.96190 

.29015 

.95698 

.30680 

.95177 

.32337 

.94627 

.33983 

.94049 

8 

53 

.27368 

.96182 

.29042 

.95690 

.30708 

.95168 

.32364 

.94618 

.34011 

.94039 

7 

54 

.27396 

.96174 

.29070 

.95681 

.30736 

.95159 

.32392 

.94609 

.34038 

94029 

6 

55 

.27424 

.96166 

.29098 

.95673 

.30763 

.95150 

.32419 

.94599 

.34065 

94019 

5 

56 

.27452 

.96158 

.291261.95664 

.30791 

.95142 

.32447 

.94590 

.34093 

94009 

4 

57 

.27480 

.96150 

.29154|.95656 

.30819 

.95133 

.32474 

.93580 

.34120 

93999 

3 

58 

.27508 

.96142 

.  29182  j.  95647 

.30846 

.95124) 

.32502 

.94571 

.34147 

93989 

2 

59 

.27536 

.96134 

.29209  L95639 

.30874 

.95115! 

.32529 

.94561 

.34175 

93979 

1 

60 

.27564 

.96126 

.29237 

.95630 

.30902 

.951061 

.32557 

.94552 

.34202 

.93969 

0 

/ 

Cosin  Sine 

Cosin 

Sine 

Cosin 

Sine  j 

Cosin 

Sine 

Cosin 

Sine 

t 

74° 

73°    1    72° 

71° 

70° 

68 


SURVEYING. 


TABLE  Ml.  —  Continued. 
NATURAL  SINES  AND  COSINES. 


33 


20° 


Sine 


Cosin 


34202 
34229 
,34257 
,34284 
,34311 
34339 
,34366 


34421 
34448 


34503 
34530 
34557 
34584 
34612 
34639 
34666 
34694 
34721 
,34748 

34775 

,34803 
34830 
,34857 
,34884 
34912 


,34993 
,35021 


,35075 
,35102 
35130 
,35157 
,35184 
35211 
35239 
,35266 


35347 
35375 
35402 
35429 
35456 
,35484 
,35511 
,35538 
,35565 

,35592 


.93919 


.93889 
.93879 


93849 


.93799 
,93789 
.93779 


.93759 
.93748 
.93738 
.93728 
.93718 
.93708 


.93077 
.93667 


21' 


Sine  Cosin 


35837 

35864 
35891 
35918 


.93337 
.93327 


359451.93316 
359731.93306 
,36000 
,36027 
,36054  .93274 


,36108 

,36135 
,36162 
,36190 


,36244 
.36271 
.36298 
.36325 
.36352 
.36379 

.36406 
.36434 
.36461 
.36488 
.36515 
.36542 
.36569 
.36596 
.36623 
.36650 


93657  .36677 
93647  |  .36704 


93637 


93616 
93606 


93585 
93575 
93565 

93555 
93544 
93534 


93514 
93503 


93472 


.35619 
,35647 
,35674 
,35701 
,35728! 
,357551 
,35782 
,35810 


93441 
93431 
93420 
93410 
93400 
93389 
93379 


Cosin  Sine  I 


.36731 
.36758 
.36785 


.36867 
.36894 


.36975 
.37002 
.37029 
.37056 
.37083 
.37110 
.37137 
.37164 
.37191 

.37218 
.37245 
.37272 
.37299 


.37353 
.37380 
.37407 
.37434 
.37461 
Cosin 


.93222 
.93211 
.93201 
.93190 
.93180 
.93169 
.93159 


.93137 
.93127 
.93116 
.93106 
.93095 
.93084 
.93074 


.93052 
.93042 


.93020 
.93010 
.92999 
.92088 
.92978 
.92967 
.92056 
.92045 
.92935 

.92924 
.92013 
.92902 
.92892 
.92881 
.92370 
.92859 
.92849 
.92338 
.92827 

.92816 
.92805 
.92794 
.92784 
.92773 
.92762 
.92751 
.92740 
.92729 
.92718 


Sine 


68° 


22° 


Sine 


.37461 
.37488 
.37515 
.37542 
.37569 
.37595 
.37622 
.37649 
.37676 
.37703 
.37730 

.37757 
.37784 
.37811 


.37865 
.37892 
.37919 
.37946 
.37973 
.37999 


.38053 
.3S080 
.38107 
.38134 
.38161 
.30188 
.38215 
.r.8241 


.38349 
.38376 
.38403 


.38456 
.38483 
.38510 


.38591 
.38617 
.38644 
.38671 


.38725 
.38752 
.38778 


.38859 
.38886 
.38912 


Cosin 
92718 
92707 
92697 
92686 
92675 
92664 
92653 
92642 
92C31 

.92G20 


.92598 
.92587 
.92576 
.92565 
.92554 
.92543 


.92521 
.92510 
.92499 


.92477 
.92466 
.92455 
.92444 
.92432 
.92421 
.92410 


.92377 


82  155 


,92332 
92321 
,92310 
,92299 
,92287 
.92276 

.92265 

.92254 
.92243 
,92231 
,92220 
92200 
92198 
92186 
92175 
92164 


.38966 
.38993; 
.39020: 
.390461 
.390731 
Oosin 


.92141 
.92130 
.92119 
.92107 
.92096 
.92085 
.92073 
.92062 
^2050 
Sine 


67° 


23C 


.92050 


SineJ  Cosin 

19073 

,39100 

,39127 

,39153 


.92028 
.92016 


391801.92005 


.39207 
.30234 
.39260 
.39287 
.39314 
.393-41 


89394 

,39421 
,39448 
,39474 
,39501 


39555 
39581 


39635 
,39661 


,39715 
,39741 

,39768 
,39795 
,39822 


,39875 


,30955 
.39982 
.40008 
.40035 
.40062 
.40088 
.40115 
.40141 

.40168 
.40195 
.40221 
.40248 
.40275 
.40301 
.40328 
.40355 
.40381 
.40408 


91994 
91982 
91971 
91959 
91948 
91936 

91925 
91914 

91902 


.91879 


91856 
91845 
91833 
91822 

91810 
91799 
91787 
91775 
91764 
91752 
91741 
91729 
,91718 
,91706 


.91671 
.91660 
.91648 
.91636 
.91625 
.91613 
.91601 
.91590 

.91578 
.91566 
.91555 
.91543 
.91531 
.91519 
.91508 
.91496 
.91484 
.91472 


.40434  .91461 
.404G1J. 91440 
.404881.91437 
.40514 -.91425 
.40541!. 91414 
.405671.91402 
.40594 '.91390 
.406211.91378 
.406471.91366 
.406741.91355 
Cosin  i  Sine 


24° 


Sine  I  Cosin 
40674^91355 
40700 j.91343  59 
,40727  .913311  58 
407531.91319  57 
40780i. 91307 j  56 
40806 i. 91295  55 


408331.91283 
40860  .91272 
91260 
91248 
91236 

91224 
91212 
91200 
91188 
91176 
91164 


.40886 
.40913 
.40939 

.40966 
.40992 
.41019 
.41045 
.41072 
.41098 
.41125 
.41151 
.41178 
.41204 

.41231 
.41257 
.41284 
.41310 
.41337 


.41390 
.41416 
.41443 
.41469 

.41496 
.41522 
.41549 
.41575 
.41602 
.41628 
.41655 
.41681 
.41707 
.41734 

.41760 

.41787 
.41813 
.41840 
.41866 
.41892 
.41919 
.41945 
.41972 
.41998 


.42051 
.42077 
.42104 
.42130 
.42156 
.42183 
.42209 


.91152:43 
.911401  42 
.91128  41 
.91116  40 

.91104!  39 
.91092 


.91080 
.91068 
.91056 
.91044 


.91020 
.91008 


.90972 
.90960 


37 
36 
85 
34 
88 
32 
31 
30 

29 
28 

XJ7 
26 
25 

.90924!  24 
.90911 |  23 


.90875 

.90863 

.90851 
.90839 


.90814 


.908021  14 
.90790!  13 


.90778 
.90766 
.90753 

.90741 
.90729 
.90717 
.90704 


,90655 
,90643 
,90631 


Cosin  Sine 
65° 


TABLES. 


69 


TABLE  VI.— Continued. 
NATURAL  SINES  AND  COSINES. 


10 


25C 


Sine 


,42262 
,42288 
,42315 
,42341 
,42367 
.42394 
,42420 
,42446 
42473 
42499 


.42552 
.42578 
.42604 
.42631 
.42657 
.42683 
.42709 
.42736 
.42762 
.42788 


.42841 


.42894 


.42946 
.42972 


.43025 
.43051 

.43077 
.43104 
.43130 
.43156 
.43182 
.43209 


.43261 

.43287 
.43313 


.43392 
.43418 
.43445 
.43471 
.43497 
.43523 
.43549 
.43575 


.43654 


.43706 
.43733 
.43759 


Cosin 


.90631 
.90618 
.90606 
.90594 
.90582 
.90569 
.90557 
.90545 
.90532 
.90520 
.90507 

.90495 
.90483 
.90470 
.90458 

90446 
.90433 
.90421 

90408 


90371 
90358 
90346 
90&34 
90321 
90309 
90296 
90234 
9,"271 
90259 


90233 
90221 
90208 
90196 
90183 
90171 
90158 
90146 
90133 

90120 

90108 
90095 
90082 
90070 
90057 
90045 
90032 
90019 
90007 


89981 
89968 


89943 
89930 


43811  .89892 


.43837 


Cosin 


.89879 


Sine 


64° 


26° 


Sine 


.43837 
.43863 


.43916 
.43942 
.43968 
.43994 
.44020 
.44046 
.44072 


.44124 
.44151 
.44177 
.44203 
.44229 
.44255 
.44281 
.44307 
.44333 


.44411 
.44437 
.44464 
.44490 
.44516 
.44542 
.44568 
.44594 
.44620 

.44646 
.44072 
.44698 
.44724 
.44750 
.44776 
.44802 
.44828 
.44854 
.44880 

.44906 
.44932 
.44958 
.44984 
.45010 
.45036 
.45062 
.45088 
.45114 
.45140 

.45166 
.-15192 
.45218 
.45243 
.45269 
.45295 
.45321 
.45347 
.45373 


Cosin 


Cosin 

789879 
.89867 
.89854 
.89841 


.89816 


.89790 
.89777 
.89764 
.89752 

.89739 
.89726 
.89713 
.89700 
.89687 
.89674 


,89610 
,89597 
,89584 
,80571 
,89558 
,89545 
,89532 
,89519 
,80506 


,83467 
,89454 
.89441 
.89428 
.89415 
.89402 


.89376 


,89350 


,89324 
,89311 


,89285 
,89272 
89259 
89245 


27° 


.89219 


.89167 
.89153 


.89127 
.89114 
.89101 
Sine 


63C 


_Sine_ 
.45399 
.45425 
.45451 
.45477 
.45503 
.45529 
.45554 
.45580 
.45606 
.45632 
.45658 

.45684 
.45710 
.45736 
.45762 
.45787 
.45813 


.45891 
.45917 

.45942 
.45968 
.45994 
.46020 
.46046 
.46072 
.46097 
.46123 
.46149 
.46175 

.46201 

.40220 
.40252 
.46278 
.40304 
.40330 
.40355 
.40381 
.40407 
.46433 

.46458 
.46454 
.46510 
.46536 
.46561 
.46587 
.46613 
.46639 
.40064 
.46690 

.46716 
.46742 
.46767 
.46793 
.46819 
.46844 
.46870 


! .46947 


i  Cosin  Sine 


Cosin 
789101 


.89074 
.89061 
.89048 
.89035 
.89021 
.89008 
.88995 


.88955 

.88942 


.88915 
.88902 
.88888 
.88875 


.88795 


.88755 
.88741 
.88728 
.88715 
.88701 


.88647 
.88034 


.88607 
.88593 
.88580 


.88539 


.88512 
.88499 
.88485 
.88472 
.88458 
.88445 
.88431 

.88417 
.88404 
.88390 
.88377 


.88308 
.88295 


62° 


28° 


Sine_ 
.46947 
.46973 


.47024 
.47050 
.47076 
.47101 
.47127 
.47153 
.47178 
.47204 

.47229 
.47255 
.47281 
.47306 
.47332 
.47358 
.47383 
.47409 
.47434 
.47460 

.47486 
.47511 
.47537 
.47562 
.47588 
.47014 
.47039 
.47605 
.47690 
.47716 

.47741 
.47767 
.47793 
.47818 
.47844 
.47869 
.47895 
.47920 
.47946 
.47971 

.47997 

.48022 
.48048 


Cosin  I 


.88254 
.88240 
.88226 


.88185 
.88172 
.88158 

.88144 
.88130 
.88117 
.88103 ' 


,88075 


,88048 
,88034 


.88006 
.87993 
.87979 
.87965 
.87951 
.87937 
.87923 
.87909 
.87896 
.87882 

.87868 

.87854 
.87840 
.87826 
.87812 
.87798 
.87784 
.87770 
.87756 
.87743 

.87729 
.87715 
.87701 


.480731.87087 
.48099  .87073 


.48124 
.48150 
.48175 

.48201 


.48277 


.48354 


.48405 
.48430 
. 48456 
.48481 


Cosin 


.87659 
.87645 
.87631 
.87617 
.87603 

.87589 
.87575 
.87561 
.87546 
.87532 
.87518 
.87504 
.87490 
.87476 
.87462 


Sine 


61' 


29C 


Sine 


.48481 


Cosin 

787462 
,_,,. 87448 

,485321.87434 
.485571.87420 
.48583  .87406 
.48608  .87391 
486341.87377 
.87363 
.87349 
.87335 
.87321 

.87306 
.87292 
.87278 
.87264 
.87250 
.87235 
.87221 
.87207 
.87193 
.87178 

.87164 
.87150 
.87136 
.87121 
.87107 
.87093 
.87079 
.87064 
.87050 


.48710 
.48735 

.48761 

.48786 
.48811 
.48837 


.48888 
.48913 
.48938 
.48964 


.49014 
.49040 
.49065 
.49090 
.49116 
.49141 
.49166 
.49192 
.49217 
.49242 


.49293 
.49318 
.49344 


.49394 
.49419 
.49445 
.49470 


.49521 
.49546 
.49571 
.49596 
.49022 
.49C47 
.49672 
.49697 
.49723 
.49748 

.49773 
.49798 
.49824 
.49849 
.49874 


.49924 
.49950 
.49975 
.50000 
Cosin 


87021 
87007 
86993 


86964 
86949 


86878 


,86791 
,86777 
,86762 


86719 
86704 


86675 


86617 
86603 


Sine 


SURVEYING. 


TABLE  VI.— Continued. 
NATURAL  SINES  AND  COSINES. 


30' 

31° 

32° 

33° 

34° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  |  Cosin 

Sine 

Cosin 

/ 

0 

.50000 

86603 

.51504 

85717 

52992 

.84805 

.54464 

.83867 

.55919 

.82904 

'60 

1 

.50025 

86588 

.51529 

85702 

53017  .84789 

.54488 

.83851 

.55943 

.828871  59 

2 

.50050 

86573 

.51554 

85687 

53041  .84774 

.54513 

.83835 

.55968 

.82871 

58 

3 

.50076 

86559 

.51579 

85672 

53066 

.84759 

.54537 

.83819 

.55992 

.82855 

57 

4 

.50101 

86544 

.51604 

85657 

.53091 

.847431 

.54561 

.83804 

.56016 

.82839  58 

5 

.50126 

86530 

.51628 

85642 

.53115 

.84728 

.54586 

.83788 

.56040 

.828221  55 

6 

.50151 

86515 

.51653 

85627 

.53140  .84712 

.546101.83772 

.56064 

.82806 

54 

7 

.50176 

86501 

.51678 

85612 

.53164 

846971 

.54635  .83756 

.56088 

.82790 

53 

8 

.50201 

86486 

.51703 

85597 

.53189 

84681' 

.546591.83740 

.56112 

.82773 

52 

9 

.50227 

86471 

.51728 

.85582 

.53214 

84666 

.  54683  i.  83724 

.56136 

.82757 

51 

10 

.50252 

86457 

.51753 

.85567 

.53238 

84650 

.54708!.  83708 

.56160 

.82741 

50 

11 

.50277 

86442 

.51778 

.85551 

.53263 

84635  T 

.54732  .83692 

.56184 

.82724 

49 

12 

.50302 

86427 

.51803 

.85536 

.53288 

84619 

.547561.83676 

.56208 

.82708 

48 

13 

.50327 

86413 

.51828 

.85521 

.53312 

84604 

.54781  .83660 

.56232 

.82692 

47 

14 

.50352 

86398 

.51852 

.85506 

.53337 

.84588 

.54805  .83645 

.56256 

.82675 

46 

15 

.50377 

86384 

.51877 

.85491 

.53361 

.84373! 

.54829  .83629 

.56280 

.82659 

45 

16 

.50403 

86369 

.51902 

.85476 

.53386 

.84557! 

.54854  .83613 

.56305 

.82643 

44 

17 

.50428 

86354 

.51927 

.85461 

.53411 

.84542! 

.54878  .83597 

.56329 

.82626 

43 

18 

.50453 

86340 

.51952 

.85446 

.53435 

.84526 

.54902 

.83581 

.56353 

.82610 

42 

19 

.50478 

86325 

.51977 

.85431 

.53460 

.84511 

.M927 

.83565 

.56377 

.82593 

41 

20 

.50503 

86310 

.52002 

.85416 

.53484 

.84495 

.54951 

.83549 

.56401 

.82577 

40 

21 

.50528 

86295 

.52026 

.85401 

.53509 

.84480 

.54975 

.83533 

.56425 

.82561 

39 

22 

.50553 

86281 

.52051 

85385 

.53534 

.84464 

.54999 

.83517 

.56449 

.82544;  38 

23 

.50578 

.86266 

.52070 

.85370 

.53558 

.84448 

.55024 

.83501 

.56473 

.82528  37 

24 

.50603 

.86251 

.52101 

.85355 

.53583 

.84433 

.55048 

.83485 

.56497 

.82511  36 

25 

.50628 

.86237 

.52126 

.85340 

.53607 

.84417 

.55072 

.83469 

.56521 

.82495  35 

26 

.50654 

.86222 

.52151 

.85325 

.53632 

.84402 

.55097 

.83453 

.56545 

.82478  34 

27 

.50679 

.86207 

.52175 

.85310 

.53656 

.84386 

.55121 

.83437 

.56569 

.82462  33 

28 

.50704 

.86192 

.52200 

.85294 

.53681 

.84370 

.55145 

.83421 

.56593 

.82446  32 

29 

.50729 

.86178 

.52225 

.85279 

.53705 

.84355 

.55169 

.83405 

.56617 

.82429:  31 

30 

.50754 

.86163 

.52250 

.85264 

.53730 

.84339 

.55194 

.83389 

.56641 

.82413 

30 

31 

.50779 

.86148 

.52275 

.85249 

.53754 

.84324 

.55218 

.83373 

.56665 

.82396 

29 

32 

.50804 

.86133 

.52293 

.85234 

.53779 

.84303 

.55242 

.83356 

.56689 

.  82380  \  28 

33 

.50829 

.86119 

.52324 

.83218 

.53804 

.84292 

.55266 

.83340 

.56713 

.82363!  27 

34 

.50854 

.861041 

.52349 

.83203 

.53828 

.84277 

.55291 

.83324 

.56736 

.823471  26 

35 

.50879 

.86089' 

.52374 

.83188 

.53853 

.84261 

.55315 

.83308 

.56760 

.82330 

25 

36 

.50904 

.86074 

.52399 

.83173 

.53877 

.84245 

.55339 

.83292 

.56784 

.82314 

24 

37 

.50929 

.86053 

.52423 

.83157 

.53902 

.84230 

.55363 

.83276 

.56808 

.82297  23 

38 

.50954 

.86045 

.52448 

.83142 

.53926 

.84214 

.55388 

.83260 

.56832 

.82281  i  22  i 

39 

.50979 

.86030 

.52473 

.83127 

.53951 

.84198 

.55412 

.83244 

.56856 

.82264,21  i 

40 

.51004 

.86015 

.52498 

.85112 

.53975 

.84182 

.55436 

.83228 

.56880 

.82248 

20 

41 

.51029 

.86000' 

.52522 

.85096 

.54000 

.84167 

.55460 

.83212 

.56904 

.82231 

19 

42 

.51054 

.85985 

.52547 

.85081 

.54024 

.84151 

.55484 

.83195 

.56928 

.822141  18 

43 

.51079 

.85970 

.52572 

.83006 

.54049 

.84135 

.55509 

.83179 

.56952 

.  82198  :  17 

44 

.51104 

.85956 

.52397 

.85051 

.54073 

.84120 

.55333 

.83163  .56976 

.82181!  18 

45 

.51129 

.85941 

.52621 

.85035 

.54097 

.84104 

.55557 

.831471  .57000 

.821651  15 

46 

.51154 

.85926 

.52646 

.85020 

.54122 

.84088 

.55581 

.831311  .57024 

.821481  14 

47 

.51179 

.85911 

.52671 

.85005 

.54146 

.84072 

.55605 

.83115 

.57047 

.82132 

13 

48 

.51204 

.85896 

.52696 

.84989 

.54171 

.84057 

.55630 

.83098 

.57071 

.82115 

12 

49 

.51229 

.85881 

.52720 

.84974 

.54195 

.84041 

.55654 

.83082 

.57095 

.82098 

11 

50 

.51354 

.85866 

.52745 

.84959 

.54220 

.84025 

.53678 

.83066 

.57119 

.82082 

10 

51 

.51279 

.85851 

.52770 

.84943 

.54244 

.84009 

.55702 

.83050 

.57143 

.82065 

9 

52 

.51304 

.85836 

.52794 

.84928 

.54269 

.83994 

.55726 

.83034 

.57167 

.82048 

8 

53 

.51329 

.85821 

.52819 

.84913 

.54293 

.83978 

.55750 

.83017 

.57191 

.82032 

7 

54 

.51354 

.85806 

.52844 

.84897 

.54317 

.83962 

.  55775 

.83001 

.57215 

.82015 

6 

55 

.51379 

.85792 

.52869 

.84882 

.54342 

.83946 

.55799 

.82985 

.57238 

.81999 

5 

56 

.51404 

.85777 

.52893 

.84866 

.54366 

.83930 

.55823 

.82969 

.57262 

.81982 

4 

57 

.51429 

.85762 

.52918 

.84851 

.54391 

.83915 

.55847 

.82953 

.57286 

.81965 

3 

58 

.51454 

.85747 

.52943 

.84836 

.54415 

.83899 

.55871 

.82936 

.57310 

.81949 

8 

59 

.51479 

.85732 

.52967 

.84820 

.54440 

.83883 

.55895 

.82920 

.57334 

.81932 

1 

60 

.51504 

.85717 

.52992 

.84805 

.54464 

.83867 

.55919 

.82904 

.57358 

.81915 

0 

/ 

Cosin 

"Siae 

Cosin 

"Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

r 

59° 

58° 

57V 

56° 

55° 

TABLES. 


TABLE  VI.  —  Contimied. 
NATURAL  SINES  AND  COSINES. 


35° 

36° 

37° 

38° 

39° 

/ 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

/ 

-Q 

.57358 

.81915 

.58779 

.80902 

.60182 

.79864 

.61566 

.78801 

.62932 

.77715-  60 

1 

.57381 

.81899 

.58802 

.80885 

.60205 

.79846 

.61589 

.78783 

.62955 

.77696  59 

2 

.57405 

.81882 

.58826 

.80867 

.60228 

.79829 

.61612 

.78765 

.62977 

.77678>  58 

3 

.57429 

.81865 

.58849 

.80850 

.60251 

.79811 

.61635 

.78747 

.63000 

.77660  57 

4 

.57453 

.81848 

.58873 

.80833 

.60274 

.79793 

.61658 

.78729 

.63022 

.776411  56 

5 

.57477 

.81832 

.58896 

.80816 

.60298 

.79776 

.61681- 

.78711 

.63045 

.77623;  55 

6 

.57501 

.81815 

.58920 

.80799! 

.60321 

.79758 

.617'04 

.78694 

.63068 

.77605  54 

7 

.57524 

.81798 

.58943 

.80782 

.60344 

.79741 

.61726 

.78676 

.63090 

.77586  53 

8 

.57548 

.81782 

.58967 

.80765 

.60367 

.79723 

.61749 

.78658 

.63113 

.77568  52 

9 

.57572 

.81765 

.58990 

.80748 

.60390 

.79706 

.61772 

.78640 

.63135 

.77550  51 

10 

.57596 

.81748 

.59014 

.80730 

.60414 

.79688 

.61795 

.78622 

.63158 

.77531  50 

11 

.57619 

.81731 

.59037 

.80713 

.60437 

.79671 

.61818 

.78604 

.63180 

.77513 

40 

12 

.57643 

.81714 

.59061 

.80696 

.60460 

.79653 

.61841 

.78586 

.63203 

.77494 

48 

13 

.57667 

.81698 

.59084 

.80679 

.60483 

.79635 

.61864 

.78568 

.632251.77476  47 

14 

.57691 

.81681 

.59108 

.80662 

.60506 

.79618 

.61887 

.78550 

.63248  .77458!  46 

15 

.57715 

.81664 

.59131 

.80644 

.60529 

.79600 

.61909  .78532 

.632711.77439 

45 

16 

.57738 

.81647 

.59154 

.80627i 

.60553 

.79583 

.61932  |.  78514  i 

.  63293  :.  77421 

44 

17 

.57762 

.81631 

.59178 

.80610 

.60576 

.79565 

.61955 

.78496 

.63316  .77402J  43 

18 

.57786 

.81614 

.59201 

.80593; 

.60599 

.79547 

.61978 

.78478 

.63338 

.77384;  42 

19 

.57810 

.81597 

.59225 

.80576 

.60622 

.79530 

.62001 

.78460 

.63361 

.77366!  41 

20 

.57833 

.81580 

.59248 

.80558 

.60645 

.79512 

.62024 

.78442 

.63383 

.77347 

40 

21 

.57857 

.81563 

.59272 

.80541 

.60668 

.79494 

.62046 

.78424 

.63406 

.77329 

39 

22 

.57881 

.81546 

.59295 

.80524 

.60691 

.79477 

.62069 

.78405 

.63428 

.77310  38 

23 

.57904 

.81530 

.59318 

.80507 

.60714 

.79459 

.62092 

.78387 

.63451 

.77292  37 

24 

.57928 

.81513 

.59342 

.80489 

.60738 

.79441 

.62115 

.78369 

.63473 

.77273  36 

25 

.57952 

.81496 

.59365 

.80472 

.60761 

.79424 

.62138 

.78351 

.63496 

.77255  35 

23 

.57976 

.81479 

.59389 

.80455! 

.60784 

.79406 

.62160 

.78333 

.63518 

.77236  34 

27 

.57999 

.81462 

.59412 

.80438! 

.60807 

.79388 

.62183 

.78315 

.63540 

.77218  33 

28 

.58023 

.81445 

.59436 

.80420 

.60830 

.79371 

.62206 

.78297 

.63563 

.77199 

32 

29 

.58047 

.81428 

.59459 

.80403 

60853 

.79353 

.62229 

.78279 

.63585 

.77181 

31 

30 

.58070 

.81412 

.59482 

.80386 

.60876 

.79335 

.62251 

.78261 

.63608 

.77162 

30 

31 

.58094 

.81395 

.59506 

.80368 

.60899 

.79318 

.62274 

.78243 

.63630 

.77144 

29 

32  .58118 

.81378 

.59529 

.80351! 

.60922 

.79300 

.62297 

.78385 

.63653 

.77125  28 

33  :  .58141 

.81361 

.59552 

.80334 

.60945 

.79282 

.62320 

.78206 

.63675  .77107,27 

34  i  .58165 

.81344 

.59576  .80316 

.60968 

.79264 

.62342 

.78188 

.63698  .77088  26 

35  .58189 

.81327 

.59599  .80299! 

.60991 

.79247 

.62365 

.78170 

.63720  .77070,  25 

36 

.58212 

.81310 

.59622 

.80282! 

.61015 

.79229 

.62388 

.78152 

.63742!.  7  7051  24 

37 

.58236 

.81293 

.59646 

.80264 

.61038 

.79211 

.62411 

.78134! 

.63765;.  77033  23 

38 

.58260 

.81276 

.59669 

.80247. 

.61061 

.79193 

.62433 

.78116| 

.63787  '.77014  22 

39 

.58383 

.81259 

.59693 

.80230 

.61084 

.79176  '1.62456 

.78098; 

.63810  .76996  21 

40 

.58307 

.81242 

.59716 

.80212 

.61107 

.79158  |  .62479 

.78079] 

.63832 

.76977  20 

41 

.58330 

.81225 

.59739 

.80195 

.61130 

.79140 

.62502 

.78061! 

.63854 

.76959  19 

42 

.58354  .81208' 

.59763 

.80178' 

.61153 

.79122 

.62524 

.78043 

.63877 

.76940  18 

43 

.58378  .811911 

.59786 

.80160 

.61176 

.79105 

.62547 

.78025 

.63899 

.76921!  17 

44 

.58401 

.81174 

.59809 

.80143 

.61199 

.79C87 

.62570 

.78007! 

.63922 

.76903  16 

45 

.58425 

.81157 

.59832 

.80125 

.61222 

.79069 

.62592 

.779881 

.63944 

.768841  15 

46 

.58449 

.81140 

.59856 

.  80108  ' 

.61245 

.79051 

.62615 

.77970 

.63966 

.76866  14 

47 

.58472 

.81123 

.59879 

.80091  ! 

.61268 

.79033 

.62638 

.  77952  ' 

.63989 

.76847 

13 

48 

.58496 

.81106 

.59902 

.  80073  ' 

.61291 

.79016 

.62660 

.779341 

.64011 

.76828 

12 

49 

.58519 

.81089 

.59926 

.80056 

.61314 

.78998 

.62683 

.77916 

.64033 

.76810 

11 

50 

.58543 

.81072 

.59949 

.  80038  | 

.61337 

.78980 

.62706 

.77897 

.64056 

.767S1 

10 

51 

.58567 

.81055 

.59972 

.80021^ 

.61360 

.78962 

.62728 

.77879 

.64078 

.76772 

9 

52 

.58590 

.810381 

.59995 

.  80003  ' 

.61383 

.78944 

.62751 

.77861 

.64100 

.76754 

8 

53 

.58614 

.81021 

.60019 

.79986, 

.61406 

.78926 

.62774 

.77843 

.64123 

.76735 

7 

54 

.58637 

.81004 

.60042 

.79968 

.61429 

.78908 

.62796 

.77824 

.64145 

.76717 

6 

55 

.58661 

.80987 

.60065 

.79951 

.61451 

.78891 

.62819 

.77806 

.64167 

.76698 

5 

56 

.58684 

.80970' 

.60089 

.79934 

.61474 

.78873 

.62842 

.77788 

.641901.76679 

4 

57 

.58708 

.  80953  i 

.60112 

.79916 

.61497 

.78855 

.62864 

.77769 

.64212  .76661 

3 

58 

.58731 

.80936 

.60136 

.79899 

.61520 

.78837 

.62887 

.77751 

.64234!.  76642 

2 

59 

.58755 

.80919 

.60158 

.79881 

.61543 

.78819 

.62909 

.77733 

.64256|.76623 

1 

60 

.58779 

.80902 

.60182 

.79864; 

.61566 

.78801 

.62932 

.77715 

.64279|.76604 

0 

e 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin  Sine 

/ 

54° 

53° 

52° 

51* 

50° 

SURVEYING. 


TABLE  Ml.— Continued. 
NATURAL  SINES  AND  COSINES. 


40° 


Sine 


,61279 
6J301 
,61323 
64346 
64368 
64390 
64412 
64435 
64457 
64479 
64501 

64524 

64546 
64568 
64590 
64612 


64657 
34679 
64701 
64723 

64746 

64768 
64790 
64812 
64834 
64856 
64878 
64901 


64945 

64967 
64989 
65011 
65033 
65055 
65077 
65100 
65122 
65144 


Cosin 


.70604 
.76586 
.76567 
.76548 
.76530 
.76511 
.76492 
.76473 
.76455 
.76438 
.76417 

.76398 


,76361 
76342 
76323! 
76304 

76286 i 
76267 
76248 


76210 
76192 
76173 
76154 
76135 
76116 
76097 
76078 
76059 
76041 

76022 
76003 
75984 
75965| 
75946 
75927 
75908 


65188 
65210 
65232 
65254 


65342 


.65430 
.65452 
.65474 
.65496 
.65518 
.65540 
.65562 
.65584 
.65606 
Cosin 


75870 
75851 


75813 
75794! 
75775! 
75756! 
75738 1 
75719 
75700 
75630. 
75661 

.75642 


.75604 
.75585 
.75566 
.75547 


.75509 
.75490 
.75471 
Sine 


49° 


41' 


Sine 


,65606 
.65628 
,65650 
,65672 
65694 
65716 
65738 
65759 
65781 
65803 


,65847 
,65869 
,65891 
,65913 
65935 
65956 
65978 
66000 
66022 
66044 


66109 
66131 
66153 
66175 
66197 
63218 
63240 


66284 
66306 
66327 
66349 
66371 
66393 
66414 
66436 
66458 
60480 


66523 
66545 
66566 
66588 
66610 


.66875 


66718 
.66740 
.66762 

.66783 


Cosin 


.75471 
.75452 
.75433 
.75414 
.75395 
.75375 
.75356 
.75337 
.75318 
.75299 
.75280 

.75261 
'5241 
.75222 
.75203 
.75184 

75165 
.75146 
.75126 
.75107 

75088 

75069 

75050 
75030 
75011 


.66848 
.66870 


Cosin 


74973 


74934 ! 
74915  i 


74876 ! 
74857 i 
74838 
74318 
74799 
74780! 
74760 1 
74741 1 
74722 
74703 

74683 
746641 
74644! 
74625 : 
7460S 
74586 1 
74567 
,74548 ! 
,74528! 
,  74509 i 


.74470 
.74451 
.74431 
.74412 
.74392 
.74373 
.74353 
.74334 
.74314 


Sine 


48° 


42° 


Sine 


,66913 


,66956 
,66978 


.67021 
.67043 
.67064 
.67086 
.67107 
.67129 

.67151 

.6717 

.67194 

.67215 

.67237 

.67258 

.67280 

.67301 

.67323 

.67344 

.67366 
.67387 
.67409 
.67430 
.67452 
.67473 
.67495 
.67516 
.67538 
.67559 

.67580 
.67602 
67623 
67645 
.67666 


67709 
67730 
67752 
,67773 

,67795 

,67816 
,67837 
.67859 


.67901 


.67944 
.67965 
.67987 

.68008 


.68051 
.68072 


.68157 
.68179 


Cosin 


Cosin 

.74314 
.74295 
.74276 
.74256 
.74237 
.74217 
.74198 
.74178 
.74159 
.74139 
.74120 

.74100 

,74080 
,74061 
,74041 
,74022 
,74002 
,73983 
,73903 
,70944 
73924 

73904 

73885 
733G5 
73840 
7382G 
7380G 
73787 
737C7 
73747 
73728 

73708 

73683 


73649 

73G29 
73G10 
73590 
73570 
73551 
73531 

73511 

73491 
73472 
73452 
73432 
73413 


73373 
73353 


73314 
73294 
.73274 
.73254 
.73234 
.73215 
.73195 
.731 
.73155 
.73135 


Sine 


47e 


43« 


Sine 
;68200 
.68221 
.68242 
.68264 
.68285 


.68327 
.68349 
.68370 
,68391 
.68413 
.68434 
.68455 
.68476 
.68497 
.68518 
.68539 
.68561 
.68582 
.CSG03 


.68645 
.63GG6 
.68688 


.68730 
.68751 

.68772 
.63793 
.63814 


,68857 

,63878 
,63399 
,63920 
,63941 
,68902 


,69004 
,69025 


,69109 


,69151 
,69172 
,69193 
.69214 


,69256 


.69445 
.t>9466 
Cosin 


Cosin 
.73135 
.73116 

.73096 
.73076 
.73056 
.73036 
.73016 
.72996 
.72976 
.72957 
.72937 

.72917 

.72897 
.72877 
.72857 
.72837 
.72817 
.72797 
.72777 
.72757 
.72737 

.72717 
.72697 
.72677 
.72G57 
.72637 
.72017 
.72597 
.72577 
.72557 
.72537 

.72517 

.72497 
.72477 
.72457 
.72437 
.72417 
.72397 
.72377 
.72357 
.72337 

.72317 
.72297 
.72277 
.72257 
.72236 
.72216 
.72196 
.72176 
.72156 
.72136 

.72116 
.72095 
.72075 
.72055 
.72035 
.72015 
.71995 
.71974 
.71954 
.71934 


Sine 


46° 


Sine 


.69487 


.69529 
.69549 
.69570 
.69591 
.69612 


.69717 
.69737 
.60758 
.69779 
.69800 
.69821 


.69904 

.69925 
.69946 
.69966 
.69987 
.70008 
.70029 
.70049 
.70070 
.70091 

.70112 

70132 
.70153 
.70174 
.70195 
.70215 


.70257 
.70277 
.70298 

.70319 
.70339 
.70360 
.70381 
.70401 
.70422 
.70443 
.70463 
.70484 
.70505 

.70525 
.70546 
.70567 
.70587 


i! |  , 

Cosin  _ 
.71934  60 
1914  59 
58 
57 
56 
55 
54 
53 
52 
51 
50 

49 

48 
47 
46 
45 


.71894 
.71873 
.71853 
.71833 
.71813 
.71792 
.71772 

,69654  .71752 
,696751.71732 

.71711 
.71691 
.71671 
.71650 
.71630 
.71610 
.71590 
.71569 
.71549 
.71529 

.71508 
.71488 
.71468 
.71447 
.71427 
.71407 
.71386 
.71366 
.71345 
.71325 

.71305 
.71284 
.71264 
.71243 
.71223 
.71203 
.71182 
.71162 
.71141 
.71121 

.71100 

.71080 


24 


.1059!  17 
71039  16 


.70628! 
.70649' 
.70670 


.70711 
Cosin 


71019 
70998 
70978 
70957 
70937 
70916 

70896 
70875 
70855 
70834 
70813 
70793 
70772 
70752 
70731 
70711 


Sine 


45a 


TABLES. 


73 


TABLE  VII. 

NATURAL  TANGENTS  AND  COTANGENTS. 


0° 

1  o                                         Oo 

3° 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.00000 

Infinite. 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811    60 

1 

.00029 

3437.75 

.01775 

56.3506 

.03521 

28.3994 

.05270 

18.9755  159 

2 

.00058 

1718.87 

.01804 

55.4415 

.03550 

28.1664 

.05299 

18.8711 

58 

3 

.00087 

1145.92 

.01833 

54.5613 

.03579 

27.9372 

.05328 

18.7678 

57 

4 

.00116 

859.436 

.01862 

53.7086 

.03609 

27.7117 

.05357 

18.6656 

56 

5 

.00145 

687.549 

.01891 

52.8821 

.03638 

27.4899 

.05387 

18.5645 

55 

6 

.00175 

572.957 

.01920 

52.0807 

.03667 

27.2715 

.05416 

18.4645 

54 

7 

.00204 

491.106 

.01949 

51.3032 

.03696 

27.0566 

.05445 

18.3655 

53 

8 

.00233 

429.718 

.01978 

50.5485 

.03725 

26.8450 

.05474 

18.2677 

52 

9 

.00262 

381.971 

.02007 

49.8157 

.037'54 

26.6367 

.05503 

18.1708 

51 

10 

.00291 

343.774 

.02036 

49.1039 

.03783 

26.4316 

.05533 

18.0750 

50 

11 

.00320 

312.521 

.02066 

48.4121 

.03812 

26.2296 

.05562 

17.9802 

49 

12 

.00349 

286.478 

.02095 

47.7395 

.03842 

26.0307 

.05591 

17.8863 

48 

13 

.00378 

264.441 

.02124 

47.0853 

.03871 

25.8348 

.05620 

17.7934 

47 

14 

.00407 

245.552 

.02153 

46.4489 

.03900 

25.6418 

.05649 

17.7015  J46 

15 

.00433 

229.182 

.02182 

45.8294 

.03929 

25.4517 

.05678 

17.6106  J45 

16 

.00465 

214.858 

.02211 

45.2261 

.03958 

25.2644 

.05708 

17.5205  144 

17 

.00495 

202.219 

.02240 

44.6386 

.03987 

25.0798 

.05737 

17.4314    43 

18 

.00524 

190.984 

.02269 

44.0G61 

.04016 

24.8978 

.05766 

17.3432    42 

19 

.00553 

180.932 

.02298 

43.5081 

.04046 

24.7185 

.05795 

17.2558 

41 

20 

.00582 

171.885 

.02328 

42.9641 

.04075 

24.5418 

.05824 

17.1693 

40 

21 

.00611 

163.700 

.02357 

42.4335 

.04104 

24.3675 

.05854 

17.0837 

39 

22 

.00640 

156.259 

.02386 

41.9158 

.04133 

24.1957 

.05883 

16.9990  138 

23 

.00669 

149.465 

.02415 

41.4106 

.04162 

24.0263 

.05912 

16.9150 

37 

21 

.00698 

143.237 

.02444 

40.9174 

.04191 

23.8593 

.05941 

16.8319 

36 

25 

.00727 

137.507 

.02473 

40.4358 

.04220 

23.6945 

.05970 

16.7496 

35 

26 

.00756 

132.219 

,02502 

39.9655 

.04250 

23.5321 

.05999 

16.6681 

34 

27 

.00785 

127.321 

.02531 

39.5059 

.04279 

23.3718 

.06029 

16.5874 

33 

88 

.00815 

122.774 

.02560 

39.0568 

.04308 

23.2137 

.06058 

16.5075 

32 

20 

.00844 

118.540 

.02589 

38.6177 

.04337 

23.0577 

.06087 

16.4283 

31 

3D 

.00873 

114.589 

.02619 

38.1885 

.04366 

22.9038 

.06116 

16.3499 

30 

31 

.00902 

110.892 

.02648 

37.7680 

.04395 

22.7519 

.06145 

16.2722 

29 

:;j 

.00931 

107.426 

.02677 

37.3579 

.04424 

22.6020 

.06175 

16.1952  128 

33 

.00960 

104.171 

.02706 

36.9560 

.04454 

22.4541 

.06204 

16.1190 

27 

84 

.00989 

101.107 

.02735 

36.5627 

.04483 

22.3081 

.06233 

16.0435 

26 

35 

.01018 

98.2179 

.02764 

36.1776 

.04512 

22.1640 

.06262 

15.9687 

25 

36 

.01047 

95.4895 

.02793 

35.8006 

.04541 

22.0217 

.06291 

15.8945 

24 

K 

.01076 

92.9085 

.02822 

35.4313 

.04570 

21.8813 

.06321 

15.8211 

23 

38 

.01105 

90.4633 

.02851 

35.0695 

.04599 

21.7426 

.06350 

15.7483 

22 

:v.i 

.01135 

88.1436 

.02881 

34.7151 

.04628 

21.6056 

.06379 

15.6762    21 

40 

.01164 

85.9398 

.02910 

34.3G78 

.04658 

21.4704 

.06408 

15.6048 

20 

41 

.01193 

83.8435 

.02939 

34.0273 

.04687 

21.3369 

.06437 

15.5340 

19 

42 

.01222 

81.8470 

.02963 

33.6935 

.04716 

21.2049 

.06467 

15.4638 

18 

43 

.01251 

79.9434 

.02997 

33.3662 

.04745 

21.0747 

.06496 

15.3943 

17 

44 

.01280 

78.1263 

.03026 

33.0452 

.04774 

20.9460 

.06525 

15.3254 

16 

45 

.01309 

76.3900 

.03055 

32.7303 

.04803 

20.8188 

.06554 

15.8571 

15 

4fi 

.C1338 

74.7292 

.03084 

32.4213 

.04833 

20.6932 

.06584 

15.1893 

14 

47 

.01367 

73.1390 

.08114 

32.1181 

.04862 

20.5691 

.06613 

15.1222 

13 

48 

.01396 

71.6151 

.03143 

31.8205 

.04891 

20.4465 

.06642 

15.0557 

12 

49 

.01425 

70.1533 

.08172 

31.5284 

.04920 

20.3253 

.06671 

14.9898 

11 

50 

.01455 

68.7501 

.03201 

31.2416 

.04949 

20.2056 

.06700 

14.9244 

10 

51 

.01484 

67.4019 

.03230 

30.9599 

.04978 

20.0872 

.06730 

14.8596 

9 

52 

.01513 

66.1055 

.03259 

30.6833 

.05007 

19.9702 

.06759 

14.7954 

8 

53 

.01542 

64.8580 

.03288 

30.4116 

.05037 

19.8546 

.06788 

14.7317 

r 

54 

.01571 

63.6567 

.03317 

30.1446 

.05066 

19.7403 

.06817 

14.6685 

6 

55 

.01600 

62.4992 

.03346 

29.8823 

.05095 

19.6273 

.06847 

14.6059 

5 

56 

.01629 

61.3829 

.03376 

29.6245 

.05124 

19.5156 

.06876 

14.5438 

4 

57 

.01658 

60.3058 

.03405 

29.3711 

.05153 

19.4051 

.06905 

14.4823 

3 

58 

.01687 

59.2659 

.03434 

29.1220 

.05182 

19.2959 

.06934 

14.4212 

2 

59 

.01716 

58.2612 

.08463 

28.8771 

.05212 

19.1879 

.06963 

14.3607 

1 

60 

.01746 

57.2900 

.03492 

28.6363 

.05241 

19.0811 

.06993 

14.3007 

0 

Cotang 

Tang 

Cotang 

Tang 

Co  tang 

Tang 

Cotang  j   Tang 

t 

89° 

88°                       87* 

86° 

74 


SUR  VE  YING. 


TABLE  VII.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


/ 

4°            I 

5° 

6° 

70 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  i 

Tang 

Cotang 

0 

.06993 

14.3007 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

60 

1 

.07022 

14.2411 

.08778 

11.3919 

.10540 

9.48781 

.12308 

8.12481 

59 

2 

.07051 

14.1821 

.08807 

11.3540 

.10569 

9.46141 

.12338 

8.10536 

58 

3 

.07080 

14.1235 

.08837 

11.3163 

.10599 

9.43515 

.12367 

8.08600 

57 

4 

.07110 

14.0655 

.08866 

11.2789 

.10628 

9.40904 

.12397 

8.06674 

56 

5 

.07139 

14.0079 

.08895 

11.2417 

.10657 

9.38307 

.15426 

8.04756 

55 

G 

.07168 

13.9507 

.08925 

11.2048 

.10687 

9.35724 

.12456 

8.02848 

54 

7 

.07197 

13.8940 

.08954 

11.1681 

.10716 

9.33155 

.12485 

8.00948 

53 

8 

.07227 

13.8378 

.08983 

11.1316 

.10746 

9.30599 

.12515 

7.99058 

52 

9 

.07256 

13.7821 

.09013 

11.0954 

.10775 

9.28058 

.12544 

7.97176 

51 

10 

.07285 

13.7267 

.09042 

11.0594 

.10805 

9.25530 

.12574 

7.95302 

50 

11 

.07314 

13.6719 

.09071 

11.0237 

.10834 

9.23016 

.12608 

7.93438 

49 

12 

.07344 

13.6174 

.09101 

10.9882 

.10863 

9.20516 

.12633 

7.91582 

48 

13 

.07373 

13.5634 

.09130 

10.9529 

.10893 

9.18028 

.12662 

7.89734 

47 

14 

.07402 

13.5098 

.09159 

10.9178 

.10922 

9.15554 

.12692 

7.87895 

46 

15 

.07431 

13.4566 

.09189 

10.8829 

.10952 

9.13093 

.12722 

7.86064 

45 

1G 

.07461 

13.4039 

.09218 

10.8483 

.10981 

9.10646 

.12751 

7.84242 

44 

17 

.07490 

13.3515 

.09247 

10.8139 

.11011 

9.08211 

.12781 

7.82428 

43 

18 

.07519 

13.2996 

.09277 

10.7797 

.11040 

9.05789 

.12810 

7.80622 

42 

19 

.07548 

13.2480 

.09306 

10.7457 

.11070 

9.03379 

.12840 

7.78825 

41 

20 

.07578 

13.1969 

.09335 

10.7119 

.11099 

9.C0983 

.12869 

7.77035 

40 

21 

.07607 

13.1461 

.09365 

10.6783 

.11128 

8.98598 

.12899 

7.75254 

39 

22 

.07636 

13.0958 

.09394 

10.6450 

.11158 

8.96227 

.12929 

7.73480 

38 

23 

.07665 

13.0458 

.09423 

10.6118 

.11187 

8.93867 

.12958 

7.71715 

37 

24 

.07695 

12.9962 

.09453 

10.5789 

.11217 

8.91520 

.12988 

7.69957 

36 

25 

.07724 

12.9469 

.09482 

10.5462 

.11246 

8.89185 

.13017 

7.68208 

35 

26 

.07753 

12.8981 

.09511 

10.5136 

.11276 

8.86862 

.13047 

7.66466 

34 

27 

.07782 

12.8496 

.09541 

10.4813 

.11305 

8.84551 

.13076 

7.64732 

33 

28 

.07812 

12.8014 

.09570 

10.4491 

.11335 

8.82252 

.13106 

7.63005 

32 

29 

.07841 

12.7536 

.09600 

10.4172 

.11364 

8.79964 

.13136 

7.61287  |31 

30 

.07870 

12.7062 

.09629 

10.3854 

.11394 

8.77689 

43165 

7.59575 

30 

81 

.07899 

12.6591 

.09658 

10.3538 

.11423 

8.75425 

.13195 

7.57872 

29 

32 

.07929 

12.6124 

.09688 

10.3224 

.11452 

8.73172 

.13224 

7.56176 

28 

33 

.07958 

12.5660 

.09717 

10.2913 

.11482 

8.70931 

.13254 

7.54487 

27 

34 

.07987 

12.5199 

.09746 

10.2602 

.11511 

8.68701 

.13284 

7.52806    26 

35 

.08017 

12.4742 

.09776 

10.2294 

.11541 

8.66482 

.13313 

7.51132    25 

36 

.08046 

12.4288 

.09805 

10.1988 

.11570 

8.64275 

.13343 

7.49465  '24 

37 

.08075 

12.3838 

.09834 

10.1683 

.11600 

8.62078 

.13372 

7.47806  |23 

38 

.08104 

12.3390 

.09864 

10.1381 

.11629 

8.59893 

.13402 

7.46154  i22 

39 

.08134 

12.2946 

.09893 

10.1080 

.11659 

8.57718 

.13432 

7.44509    21 

40 

.08163 

12.2505 

.09923 

10.0780 

.11688 

8.55555 

.13461 

7.42871 

20 

41 

.08192 

12.2067 

.09952 

10.0483 

.11718 

f  53402 

.13491 

7.41240 

19 

42 

.08221 

12.1632 

.09981 

10.0187 

.11747 

8.51259 

.13521 

7.39616 

18 

43 

.08251 

12.1201 

.10011 

9.98931 

.11777 

8.49128 

.13550 

7.37999 

17 

44 

.08280 

12.0772 

.10040 

9.96007 

.11806 

8.47007 

.13580 

7.36389 

16 

45 

.08309 

12.0346 

.10069 

9.93101 

.11836 

8.44896 

.13609 

7.34786 

15 

46 

.08339 

11.9923 

.10099 

9.90211 

.11865 

8.42795 

.13639 

7.33190 

14 

47 

.08368 

11.9504 

.10128 

9.87338 

.11895 

8.40705 

.13669 

7.31600 

13 

48 

.08397 

11.9087 

.10158 

9.84482 

.11924 

8.38625 

.13698 

7.30018 

12 

49 

.08427 

11.8673 

.10187 

9.81641 

.11954 

8.36555 

.13728 

7.28442 

11 

50 

.08456 

11.8262 

.10216 

9.78817 

.11983 

8.34496 

.13758 

7.26873 

10 

51 

.08485 

11.7853 

.10246 

8.76009 

.12013 

8.32446 

.13787 

7.25310 

9 

52 

.08514 

11.7448 

.10275 

9.73217 

.12042 

8.30406 

.13817 

7.23754 

8 

53 

.08544 

11.7045 

.10305 

9.70441 

.12072 

8.28376 

.13846 

7.22204 

7 

54 

.08573 

11.6645 

.10334 

9.67680 

.12101 

8.26355 

.13876 

7.20661 

6 

55 

.08602 

11.6248 

.10363 

9.64935 

.12131 

8.24345 

.13906 

7J9125 

5 

56 

.08632 

11.5853 

.10393 

9.62205 

.12160 

8.22344 

.13935 

7.17594 

4 

57 

.08661 

11.5461 

.10422 

9.59490 

.12190 

8.20352 

.13965 

7.16071 

3 

58 

.08690 

11.5072 

.10452 

9.56791 

.12219 

8.18370 

.13995 

7.14553 

2 

59 

.08720 

11.4685 

.10481 

9.54106 

.12249 

8.16398 

.14024 

7.13042 

1 

60 

.08749 

11.4301 

.10510 

9.51436 

.12278 

8.14435 

.14054 

7.11537 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

1 

85° 

84° 

83° 

82° 

TABLES. 


75 


TABLE  Vll.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


8° 

9° 

10° 

11° 

t 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

o 

.14054 

7.11537 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

60 

1 

.14084 

7.10038 

.15868 

6.30189 

.17663 

5.66165 

.19468 

5.13658 

59 

2 

.14113 

7.08546 

.15898 

6.29007 

.17693 

5.65205 

.19498 

5.12862 

58 

a 

.14143 

7.07059 

.15928 

6.27829 

.17723 

5.64248 

.19529 

5.12069 

57 

4 

.14173 

7.05579 

.15958 

6.26655 

.17753 

5.63295 

.19559 

5.11279 

5G 

5 

.14202 

7.04105 

.15988 

6.25486 

.17783 

5.62344 

.19589 

5.10490 

55 

6 

.14232 

7.02637 

.16017 

6.24321 

.17813 

5.61397 

.19619 

5.09704 

5-1 

7 

.14262 

6.91174 

.16047 

6.23160 

.17843 

5.60452 

.19649 

5.08921 

53 

8 

.14291 

6.99718 

.16077 

6.22003 

.17873 

5.59511 

.19680 

5.08139 

52 

9 

.14321 

6.98268 

.16107 

6.20851 

.17903 

5.58573 

.19710 

5.07360 

51 

10 

.14351 

6.96823 

.16137 

6.19703 

.17933 

5.57638 

.19740 

5.06584 

5d 

11 

.14381 

6.95385 

.16167 

6.18559 

.17963 

5.56706 

.19770 

5.05809 

49 

12 

.14410 

6.93952 

.16196 

6.17419 

.17993 

5.55777 

.19801 

5.05037 

48 

13 

.14440 

6.92525 

.16226 

6.16283 

.18023 

5.54851 

.19831 

5.042G7 

47 

14 

.14470 

6.91104 

.16256 

6.15151 

.18053 

5.53927 

.19861 

5.03499 

46 

15 

.14499 

6.89688 

.16286 

6.14023 

.18083 

5.53007 

.19891 

5.02734 

45 

16 

.14529 

6.88278 

.16316 

6.12899 

.18113 

5.52090 

.19921 

5.01971 

44 

17 

.14559 

6.86874 

.16346 

6.11779 

.18143 

5.51176 

.19952 

5.01210 

43 

18 

.14588 

6.85475 

.16376 

6.10664 

.18173 

5.50264 

.19982 

5.00451 

42 

19 

.14618 

6.84082 

.16405 

6.09552 

.18203 

5.49C56 

.20012 

4.99695 

41 

20 

.14643 

6.82694 

.16435 

6.08444 

.18233 

5.48451 

.20042 

4.98940 

40 

21 

.14678 

6.81312 

.16465 

6.07340 

.18263 

5.47548 

.20073 

4.98188 

39 

22 

.14707 

6.79936 

.16495 

6.06240 

.18293 

5.4GG48 

.20103 

4.97438 

38 

23 

.14737 

6.78564 

.16525 

6.05143 

.18323 

5.45751 

.20133 

4.96690 

37 

24 

.14767 

6.77199 

.16555 

6.04031 

.18353 

5.44857 

.20164 

4.95945 

36 

25 

.14796 

6.75838 

.16585 

6.02962 

.18384 

5.43966 

.20194 

4.95201 

35 

26 

.14826 

6.74483 

.16615 

6.01878 

.18414 

5.43077 

.20224 

4.94460 

34 

27 

.14856 

6.73133 

.16645 

6.00797 

.18444 

5.42192 

.20254 

4.93721 

33 

28 

.14886 

6.71789 

.16674 

5.99720 

.18474 

5.41309 

.20285 

4.92984 

83 

29 

.14915 

6.70450 

.16704 

5.93646 

.18504 

5.40429 

.20315 

4.92249 

31 

30 

.14945 

6.69116 

.16734 

5.97576 

.18534 

5.39552 

.20345 

4.91516 

30 

31 

.14975 

6.67787 

.16764 

5.96510 

.18564 

5.38677 

.20376 

4.90785 

29 

32 

.15005 

6.66463 

.16794 

5.95448 

.18594 

5.37805 

.20406 

4.90056 

28 

33 

.15034 

6.65144 

.16824 

5.94390 

.18624 

5.36936 

.20436 

4.89330 

27 

34 

.15064 

6.63831 

.16854 

5.93335 

.18654 

5.36070 

.20466 

4.88605 

26 

35 

.15094 

6.62523 

.16884 

5.92283 

.18684 

5.35206 

.20497 

4.878S2 

25 

36 

.15124 

6.61219 

.16914 

5.91236 

.18714 

5.34345 

.20527 

4.87162 

24 

37 

.15153 

6.59921 

.16944 

5.90191 

.18745 

5.33487 

.20557 

4.86444 

23 

38 

.15183 

6.58627 

.16974 

5.89151 

.18775 

5.32631 

.20588 

4.85727 

22 

39 

.15213 

6.57339 

.17034 

5.88114 

.18805 

5.31778 

.20618 

4.85013 

21 

40 

.15243 

6.56055 

.17033 

5.87080 

.18835 

5.30928 

.20648 

'4.84300 

20 

41 

.15272 

6.54777 

.17063 

5.86051 

.18865 

5.30080 

.20679 

4.83590 

19 

42 

.15302 

6.53503 

.17093 

5.85024 

.18895 

5.29235 

.20709 

4.82882 

18 

43 

.15332 

6.52234 

.17123 

5.84001 

.18925 

5.28393 

.20739 

4.82175 

17 

44 

.15362 

6.50970 

.17153 

5.82982 

.18955 

5.27553 

.20770 

4.81471 

16 

45 

.15391 

6.49710 

.17183 

5.81966 

.18986 

5.26715 

.20800 

4.80769 

15 

46 

.15421 

6.48456 

.17213 

5.80953 

.19016 

5.25880 

.20830 

4.80068 

14 

47 

.15451 

6.47206 

.17243 

5.79944 

.19046 

5.25048 

.20861 

4.79370 

13 

48 

.15481 

6.45961 

.17273 

5.78938 

.19076 

5.24218 

.20891 

4.78673 

12 

49 

.15511 

6.44720 

.17303 

5.77936 

.19106 

5.23391 

.20921 

4.77978 

11 

50 

.15540 

6.43484 

.17333 

5.76937 

.19136 

5.22566 

.20952 

4.77386 

10 

51 

.15570 

6.42253 

.17363 

5.75941 

.19166 

5.21744 

.20982 

4.76595 

9 

52 

.15600 

6.41026 

.17393 

5.74949 

.19197 

5.20925 

.21013 

4.75906 

8 

53 

.15630 

6.39804 

.17423 

5.73960 

.19227 

5.20107 

.21043 

4.75219 

7 

54 

.15660 

6.38587 

.17453 

5.72974 

.19257 

5.19293 

.21073 

4.74534 

6 

55 

.15689 

6.37374 

.17483 

5.71992 

.19287 

5.18480 

.21104 

4.73851 

5 

56 

.15719 

6.36165 

.17513 

5.71013 

.19317 

5.17671 

.21134 

4.73170 

4 

57 

.15749 

6.34961 

.17543 

5.70037 

.19347 

5.16863 

.21164 

4.72490 

3 

58 

.15779 

6.33761 

.17573 

5.69064 

.19378 

5.16058 

.21195 

4.71813 

2 

59 

.15809 

6.32566 

.17603 

5.68094 

.19408 

5.15256 

.21225 

4.71137 

1 

60 

.15838 

6.31375 

.17633 

5.67128 

.19438 

5.14455 

.21256 

4.70463 

_0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

L 

81° 

80° 

79° 

78° 

SURVEYING. 


TABLE  V\\.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


12° 

13°                       143 

15° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

0 

.21256 

4.70463 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

60 

1 

.21286 

4.69791 

.33117 

4.32573 

.24964 

4.00582 

.26826 

3.72771 

59 

2 

.21316 

4.69121 

.23148 

4.32001 

.24995 

4.00086 

.26857 

3.72338 

58 

3 

.21347 

4.68452 

.23179 

4.31430 

.25026 

3.99592 

.26888 

3.71907 

57 

4 

.21377 

4.67786 

.23209 

4.30860 

.25056 

3.99099 

.26920 

3.71476 

56 

5 

.21408 

4,67121 

.23240 

4.30291 

.25087 

3.98607 

.26951 

3.71046 

55 

6 

.21438 

4.66458 

.23271 

4.29724 

.25118 

3.98117 

.26982 

3.70616 

M 

.21469 

4.65797 

.23301 

4.29159 

.25149 

3.97627 

.27013 

3.70188 

53 

g 

.21499 

4.65138 

.23332 

4.28595 

.25180 

3.97139 

.27CM4 

3.69761 

52 

9 

.8U5S9 

4.64480 

.23363 

4.28032 

.25211 

3.96651 

.27076 

3.69335 

51 

10 

.21560 

4.63825 

.23393 

4.27471 

.25242 

3.96165 

.27107 

3.68909 

50 

11 

.21590 

4.63171 

.23424 

4.26911 

.25273 

3.95680 

.27138 

3.68485 

49 

l| 

.21621 

4.62518 

.23455 

4.26352 

.25304 

3.95196 

.27169 

3.68061 

48 

13 

.21651 

4.61868 

.23485 

4.25795 

.25335 

3.94713 

.27201 

3.67638 

47 

M 

.21682 

4.61219 

.23516 

4.25239 

.25366 

3.94232 

.27232 

3.67217 

46 

15 

.21712 

4.60572 

.23547 

4.24685 

.25397 

3.93751 

.27263 

3.66796 

45 

16 

.21743 

4.59927 

.2357'8 

4.24132 

.25428 

3.93271 

.27294 

3.66876 

4-1 

17 

.21773 

4.59283 

.23608 

4.23580 

.25459 

3.92793 

.27326 

3.65957 

43 

18 

.21804 

4.58641 

.23639 

4.23030 

.25490 

3.92316 

.27357 

3.65538 

48 

19 

.21834 

4.58001 

.23670 

4.22481 

.25521 

3.91839 

.27388 

3.65121 

41 

20 

.21864 

4.57363 

.23700 

4.21933 

.25553 

3.91364 

.27419 

3.64705 

40 

21 

.21895 

4.56726 

.23731 

4.21387 

.25583 

3.90890 

.27451 

3.64289 

39 

2-2 

.21925 

4.56091 

.23762 

4.20842 

.25614 

3.90417 

.27482 

3.63874 

as 

23 

.21956 

4.55458 

.23793 

4.20298 

.25645 

3.89945 

.27'513 

3.63461 

37 

£4 

.21986 

4.54826 

.23823 

4.19756 

.25676 

3.89474 

.27545 

3.63048 

3(3 

25 

.22017 

4.54196 

.23854 

4.19215 

.25707 

3.89004 

.27576 

3.62636 

35 

26 

.22047 

4.53568 

.23885 

4.18675 

.25738 

3.88536 

.27607 

3.62224 

34 

27 

.22078 

4.b2941 

.23916 

4.18137 

.25769 

3.88068 

.27638 

3.61814 

88 

28 

.22108 

4.52316 

.23946 

4.17600 

.25800 

3.87601 

.27670 

3.61405 

32 

29 

.22139 

4.51693 

.23977 

4.17064 

.25831 

3.87136 

.27701 

3.60996 

31 

30 

.22169 

4.51071 

.24008 

4.16530 

.25862 

3.86671 

.27732 

360588 

30 

31 

.22200 

4.50451 

.24039 

4.15997 

.25893 

3.86208 

.27764 

3.60181 

29 

32 

.22231 

4.49832 

.24069 

4.15465 

.25924 

3.85745 

.27795 

3.59775 

2*-? 

33 

.22261 

4.49215 

.24100 

4.14934 

.25955 

3.85284 

.27826 

3.59370 

27 

84 

.22292 

4.48600 

.24131 

4.14405 

.25986 

3.84824 

.27858 

3.58966 

26 

35 

.22322 

4.47986 

.24162 

4.13877 

.26017 

3.84364 

.27889 

3.58562 

25 

30 

.22353 

4.47374 

.24193 

4.13350 

.26048 

3.83906 

.27921 

3.58160 

24 

37 

.22383 

4.46764 

.24223 

4.12825 

.26079 

3.83449 

.27952 

3.57758 

23 

38 

.22414 

4.46155 

.24254 

4.12301 

.26110 

3.82992 

.27983 

3.57357 

23 

39 

.22444 

4.45548 

.24285 

4.11778 

.26141 

3.82537 

.28015 

3.56957 

21 

40 

.22475 

4.44942 

.24316 

4.11256 

.26172 

3.82083 

.28046 

3.56557 

20 

41 

.22505 

4.44338 

.24347 

4.10736 

.26203 

3.81630 

.28077 

3.56159 

10 

42 

.22536 

4.43735 

.24377 

4.10216 

.26235 

3.81177 

.28109 

3.55761 

18 

13 

.22567 

4.43134 

.24408 

4.09699 

.26286 

3.80726 

.28140 

3.55364 

17 

44 

.22597 

4.42534 

.24439 

4.09182 

.26297 

3.80276 

.28172 

3.54968 

1(5 

45 

.22628 

4.41936 

.24470 

4.08666 

.26328 

3.79827 

.28203 

3.54573 

15 

46 

.22658 

4.41340 

.24501 

4.08152 

.26359 

3.79378 

.28234 

3.54179 

14 

47 

.22689 

4.40745 

.24532 

4.07639 

.26390 

3.78931 

.28266 

3.53785 

13 

48 

.22719 

4.40152 

.24562 

4.07127 

.26421 

3.78485 

.28297 

3.53393 

12 

49 

.22750 

4.39560 

.24593 

4.06616 

.26452 

3.78040 

.28329 

3.53001 

11 

50 

.22781 

4.38969 

.24624 

4.06107 

.26483 

3.77595 

.28360 

3.52609 

10 

51 

.22811 

4.38381 

.24655 

4.05599 

.26515 

3.77152 

.28391 

3.52219 

9 

52 

.22842 

4.37793 

.24686 

4.05092 

.26546 

3.76709 

.28423 

3.51829 

8 

53 

.22872 

4.37207 

.24717 

4.04586 

.26577 

3.76268 

.28454 

3.51441 

7 

54 

.22903 

4.36623 

.24747 

4.04081 

.26608 

3.75828 

.28486 

3.51053 

e 

55 

.22934 

4.36040 

.24778 

4.03578 

.26639 

3.75388 

.28517 

3.50666 

5 

56 

.22964 

4.35459 

.24809 

4.03076 

.26670 

8.74950 

.28549 

3.50279 

4 

57 

.22995 

4.34879 

.24840 

4.02574 

.26701 

3.74512 

.28580 

3.49894 

3 

58 

.23026 

4.34300 

.24871 

4.02074 

.26733 

3.74075 

.28612 

3.49509 

2 

59 

.23056 

4.33723 

.24902 

4.01576 

.25764 

3.73640 

.28643 

3.49125 

1 

60 

.23087 

4.33148 

.24933 

4.01078 

.26795 

3.73205 

.28675 

8.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang     Tang 

/ 

77° 

76°          II          75° 

74 

TABLES. 


77 


TABLE  VII.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


31 


16° 


.28675 
.26706 
.28738 
.28769 
.28800 


.28864 
.28895 
.28927 


.28990 

.29021 
.29053 
.29084 
.29116 
.29147 
.29179 
.29210 
.29242 
.29274 
.29305 


.29400 
.29432 
.29403 
.29495 
.29526 
.29558 
.29590 
.29621 


.29685 
.29716 
.29748 
.29780 
.29811 
.29843 
.29875 
.29906 
.29938 

.29970 
.30001 
.30033 
.30065 
.30097 
.30128 
.30160 


.30224 
.30255 

.30287 
.30319 
.30351 


.30414 
.30446 
.30478 


.30541 

30573^ 

Cotang 


Cotang 


3.48741 
3.48359 
3.47977 
3.47596 
3.47216 
3.46837 
3.46458 
3.46080 
3.45703 
3.45327 
3.44951 

3.44576 
3.44202 
3.43829 
3.43456 
3.43084 
3.42713 
3.42343 
3.41973 
3.41604 
3.41236 


3.40502 
3.40136 
3.39771 
3.39406 
3.39042 
3.38679 
3.38317 
3.37955 
3.37594 

3.37234 
3.36875 
3.36516 
3.36158 
3.35800 
3.35443 
3.3508? 
3.34733 
3.34377 
3.34023 

3.33670 
3.33317 
3.32965 
3.32614 
3.32264 
3.31914 
3.31565 
3.31216 


3.30521 

3.30174 
3.29829 
3.29483 
3.29139 
3.28795 
3.28452 
3.28109 
3.27767 
8.27426 
3.27085 


Tang 


73° 


17C 


Tang 

.30573 
.30605 
.30637 


.30700 
.30732 
.30764 
.30796 


.30891 

.30923 
.30955 
.30987 
.31019 
.31051 
.31083 
.31115 
.31147 
.31178 
.31210 


.31274 
.31306 
.31338 
.31370 
.31402 
.31434 
.31466 
.31498 
.31530 

.31563 
.31594 


.31658 
.31690 
.31722 
.31754 
.31786 
.31818 
.31850 

.31882 
.31914 
.31946 
.31978 
.32010 
.32042 
.32074 
.32106 
.32139 
.32171 

.32203 


.32267 


.32363 
.32396 
.32428 


.32492 


Cotang 


Tang 


72° 


18° 


Tang 


.32524 
.32558 


.32653 
.32685 
.32717 
.32749 

.32782 


.32878 
.32911 
.32943 
.32975 
.33007 
.33040 
.33072 
.33104 
.33136 

.33169 
.33201 
.33233 
.33200 
.33298 
.£3330 
.33363 
.83395 
.83427 
.23460 

.33492 
.33524 
.33557 
.33589 
.33621 
.33654 
.33686 
.33718 
.33751 
.33783 


.33913 
.33945 
.33978 
.34010 
.34043 
.34075 
.34108 

.34140 
.34173 
.34205 
.34238 
.34270 
.34303 


.34400 
.34433 


Cotang 


Cotang 


71° 


19° 


Tang 


.34465 


.34530 
.34563 


.34661 


.34726 
.34758 

.34791 


.34856 


.34922 
.34954 
.34987 
.35020 
.35052 
.35085 

.35118 
.35150 
.35183 
.35216 


.35314 
.35346 
.35379 
.35412 

.35445 
.35477 
.35510 
.35543 
.35576 
.35608 
.35641 
.35674 
.35707 
.35740 

.35772 


.35871 
.35904 
.35937 


.36002 
.36035 
.36068 

.36101 
.36134 
.36167 


.36265 
.36298 
.36331 


Cotang 


2.90421 
2.90147 


2.89055 
2.88783 
2.88511 
2.88240 
2.87970 
2  87700 

2.87430 
2.871(31 


2.86624 
2.86356 


2.85822 
2.85555 
2.85289 
2.85023 

2.84758 
2.84494 
2.84229 


2.83702 
2.83439 
2.83176 
2.82914 
2.82653 
2.82391 

2.82130 
2.81870 
2.81610 
2.81350 
2.81091 
2.80833 
2.80574 


2.79545 

2. 

2. 

2.78778 
2.78523 


2.78014 
2.77761 
2.77507 
2.77254 

2.77002 
2.76750 
2.76498 
2.76247 
2.75996 
2.75746 
2.75496 
2.75246 
2.74997 
2.74748 


Cotang     Tang 
70° 


37 


31 


SURVEYING. 


TABLE  VII.—  Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


/ 

20° 

21° 

22° 

23° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

/ 

~o 

.36397 

2.74748 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

60 

] 

.36430 

2.74499 

.38420 

2.60283 

.40436 

2.47302 

.42482 

2.35395 

59 

j 

.36463 

2.74251 

.38453 

2.60057 

.40470 

2.47095 

.42516 

2.35205 

58 

1 

.36496 

2.74004 

.38487 

2.59831 

.40504 

2.46888 

.42551 

2.35015 

57 

t 

.36529 

2.73756 

.38520 

2.59606 

.40538 

2.46682 

.42585 

2.34825 

56 

5 

.36562 

2.73509 

.38553 

2.59381 

.40572 

2.46476 

.42619 

2.34636 

55 

G 

.36595 

2.73263 

.38587 

2.59156 

.40606 

2.46270 

.42654 

2.34447 

54 

\ 

.36628 

2.73017 

.38620 

2.58932 

.40640 

2.46065 

.42688 

2.34258 

53 

8 

.36661 

2.72771 

.38654 

2.58708 

.40674 

2.45860 

.42722 

2.34069 

52 

9 

.36694 

2.72526 

.38687 

2.58484 

.40707 

2.45655 

.42757 

2.33881 

51 

10 

.36727 

2.72281 

.38721 

2.58261 

1  .40741 

2.45451 

.42791 

2.33693 

50 

11 

.36760 

2.72036 

.38754 

2.58038 

.40775 

2.45246 

.42826 

2.33505 

49 

12 

.36793 

2.71792 

.38787 

2.57815 

1  .40809 

2.45043 

.42860 

2.33317 

48 

13 

.36826 

2.71548 

.38821 

2.57593 

.40843 

2.44839 

.42894 

2.33130 

47 

14 

.36859 

2.71305 

.38854 

2.57371 

.40877 

2.44636 

.42929 

2.32943 

46 

15 

.36892 

2.71062 

.38888 

2.57150 

.40911 

2.44433 

.42963 

2.32756 

45 

1G 

.36925 

2.70819 

.38921 

2.56928 

.40945 

2.44230 

.42998 

2.32570 

44 

17 

.36958 

2.70577 

.38955 

2.56707 

.40979 

2.44027 

.43032 

2.32383 

43 

IB 

.36991 

2.70335 

.88988 

2.56487 

.41013 

2.43825 

.43067 

2.32197 

42 

19 

.37024 

2.70094 

.39022 

2.56266 

.41047 

2.43623 

.43101 

2.32012 

41 

2L 

.37057 

2.69853 

.39055 

2.56046 

.41081 

2.43422 

.43136 

2.31826 

40 

21 

.37090 

2.69612 

.39089 

2.55827 

.41115 

2.43220 

.43170 

2.31641 

39 

22 

.37123 

2.69371 

.39122 

2.55608 

.41149 

2.43019 

.43205 

2.31456 

38 

23 

.37157 

2.69131 

.39156 

2.55389 

.41183 

2.42819 

.43233 

2.31271 

37 

24 

.37190 

2.68892 

.39190 

2.55170 

.41217 

2.42G18 

.43274 

2.31086 

36 

25 

.37223 

2.68653 

.39223 

2.54952 

.41251 

2.42418 

.43308 

2.30902  135 

26 

.37256 

2.68414 

.39257 

2.54734 

.41285 

2.42218 

.43343 

2.30718 

34 

27 

.37289 

2.68175 

.39290 

2.54516 

.41319 

2.42019 

.43378 

2.30534 

33 

28 

.37322 

2.67937 

.39324 

2.54299 

.41353 

2.41819 

.43412 

2.30351 

32 

29 

.37355 

2.67700 

.39357 

2.54082 

.41387 

2.41620 

.43447 

2.30167 

31 

30 

.37388 

2.67462 

.39391 

2.53865 

.41421 

2.41421 

.43481 

2.29984 

30 

31 

.37422 

2.67225 

.39425. 

2.53648 

.41455 

2.41223 

.43516 

2.29801 

29 

32 

.37455 

2.66989 

.39453 

2.53432 

.41490 

2.41025 

.43550 

2.29619 

28 

33 

.37488 

2.66752 

.33492 

2.53217 

.41524 

2.40827 

.43585 

2.29437 

27 

34 

.37521 

2.66516 

.39526 

2.53001 

.41558 

2.40629 

.43620 

2.29254 

26 

35 

.37554 

2.66281 

.39559 

2.52786 

.41592 

2.40432 

.43654 

2.29073 

25 

315 

.37588 

2.66046 

.39593 

2.52571 

.41626 

2.40235 

.43689 

2.28891  |24 

37 

.37621 

2.65811 

.39626 

2.52357 

.41660 

2.40038 

.43724 

2.28710  |23 

38 

.37654 

2.65576 

.39660 

2.52142 

.41694 

2.39841 

.43758 

2.28528  '22 

39 

.37687 

2.65342 

.39G91 

2.51929 

.41728 

2.39645 

.43793 

2.28348  !21 

40 

.37720 

2.65109 

.39727 

2.51715 

.41763 

2.39449 

.43828 

2.28167 

20 

41 

.37754 

2.64875 

.39761 

2.51502 

.41797 

2.39253 

.43862 

2.27987 

19 

42 

.37787 

2.64642 

.39795 

2.51289 

.41831 

2.39058 

.43897 

2.27806    18 

43 

.37820 

2.64410 

.39829 

2.51076 

.41805 

2.38863 

.43932 

2.27626    17 

41 

.37853 

2.64177 

.39862 

2.50864 

41893 

2.38668 

.43966 

2.27447  |16 

45 

.37887 

2.63945 

.39896 

2.50652 

.41933 

2.38473 

.44001 

2.27267 

15 

40 

.37920 

2.63714 

.39930 

2.50440 

.41968 

2.38279 

.44036 

2  27088 

14 

47 

.37953 

2.63483 

.39963 

2.50229 

.42002 

2.380S4 

.44071 

2.2G909 

13 

48 

.37986 

2.63252 

.39997 

2.50018 

.42036 

2.37891 

.44105 

2.26730 

12 

49 

.38020 

2.63021 

.40031 

2.49807 

.42070 

2.37G97 

.44140 

2.26552 

11 

50 

.38053 

2.62791 

.40065 

2.49597 

.42105 

2.37504 

.44175 

2.26374 

10 

51 

.38086 

2.62561 

.40098 

2.49386 

.42139 

2.37311 

.44210 

2.26196 

9 

52 

.38120 

2.62332 

.40132 

2.49177 

.42173 

2.37118 

.44244 

2.26018 

8 

53 

.38153 

2.62103 

.40166 

2.48967 

.42207 

2.36925 

.44279 

2.25840 

7 

54 

.38186 

2.61874 

.40200 

2.48758 

.42242 

2.36733 

.44314 

2.25663 

6 

55 

.38220 

2.61646  1 

.40234  |  2.48549 

.42276 

2.36541 

.44349 

2.25486 

5 

56 

.38253 

2.61418 

.40267 

2  48340 

.42310 

2.36349 

.44384 

2.25309 

4 

57 

.38286 

2.61190 

.40301 

2.48132 

.42345 

2.36158 

.44418 

2.25132 

3 

58 

.38320 

2.60963 

.40335 

2.47924 

.42379 

2.35967 

.44453 

2.24956 

2 

59 

.38353 

2.60736 

.40369 

2.47716 

.42413 

2.35776 

.44488 

2.24780 

1 

GO 

.38386 

2.60509 

.40403 

2.47509 

.42447 

2.35585 

.44523 

2.24604 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

69°           1 

68°                       67° 

60° 

TABLES. 


79 


TABLE  VII.  —  Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


2 

4o 

2 

5° 

2 

6° 

2 

7° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~0 

.44523 

2.24604 

.46631 

2.14451 

'•   .48773 

2.05030 

.50953 

1.96261 

60 

1 

.44558 

2.24428 

.46666 

2.14288 

.48809 

2.04879 

.50989 

.96120 

59 

2 

.44593 

2.24252 

.46702 

2.14125 

.48845 

2.04728 

.51026 

.95979 

58 

3 

.44627 

2.24077 

.46737 

2.13963 

.48881 

2.04577 

.51063 

.95838 

57 

4 

.44662 

2.23902 

.46772 

2.13801 

.48917 

2.04426 

.51099 

.95698 

56 

5 

.44697 

2.23727 

.46808 

2.13639 

.48953 

2.04276 

.51136 

.95557 

55 

6 

.44732 

2.23553 

.46843 

2.1.3477 

.48989 

2.04125 

.51173 

.95417 

54 

7 

.44767 

2.23378 

.46879 

2.13316 

.49026 

2.03975 

.51209 

.95277 

53 

8 

.44802 

2.23204 

.46914 

2.13154 

.49062 

2.03825 

.51246 

.95137 

52 

9 

.44837 

2.23030 

.46950 

2.12993 

.49098 

2.03675 

.51283 

.94997 

51 

10 

.44872 

2.22857 

.46985 

2.12833 

.49134 

2.03526 

.51319 

.94858 

50 

11 

.44907 

2.22683 

.47021 

2.12671 

.49170 

2.03376 

.51356 

.94718 

49 

id 

.44942 

2.22510 

.47056 

2.12511 

.49206 

2.03227 

.51393 

.94579 

48 

13 

.44977 

2.22337 

.47092 

2.12350 

.49242 

2.03078 

.51430 

.94440 

47 

14 

.45012 

2.22164 

.47128 

2.12190 

.49278 

2.02929 

.51467 

.94301 

46 

15 

.45047 

2.21992 

.47163 

2.12030 

.49315 

2.02780 

.51503 

.94162 

45 

16 

.45082 

2.21819 

.47199 

2.11871 

.49351 

2.02631 

.51540 

.94023 

44 

17 

.45117 

2.21647 

.47234 

2.11711 

.49387 

2.02483 

.51577 

.93885 

43 

18 

.45152 

2.21475 

.47270 

2.11552 

.49423 

2.02335 

.51614 

.93746 

42 

to 

.45187 

2.21304 

.47305 

2.11392 

.49459 

2.02187 

.51651 

.93608 

41 

20 

.45222 

2.21132 

.47341 

2.11233 

.49495 

2.02039 

.51688 

.93470 

40 

21 

.45257 

2.20961 

.47377 

2.11075 

.49532 

2.01891 

.51724 

.93332 

39 

22 

.45292 

2.20790 

.47412 

2.10916 

.49568 

2.01743 

.51761 

.93195 

38 

23 

.45327 

2.20619 

.47448 

2.10758 

.49604 

2.01596 

.51798 

.93057 

37 

24 

.45362 

2.20449 

.47483 

2.10600 

.49640 

2.01449 

.51835 

.92920 

36 

25 

.45397 

2.20278 

.47519 

2.10442 

.49677 

2.01302 

.51872 

.92782 

35 

20 

.45432 

2.20108 

.47555 

2.10284 

.49713 

2.01155 

.51909 

.92645 

34 

27 

.45467 

2.19938 

.47590 

2.10126 

.49749 

2.01008 

.51946 

.92508 

33 

28 

.45502 

2.19769 

.47626 

2.09969 

.49786 

2.00862 

.51983 

.92371 

32 

29 

.45538 

2.19599 

.47662 

2.09811 

.49822 

2.00715 

.52020 

.92235 

31 

30 

.45573 

2.19430 

.47698 

2.09654 

.49858 

2.00569 

.52057 

.92098 

30 

31 

.45608 

2.19261 

.47733 

2.09498 

.49894 

2.00423 

.52094 

.91962 

29 

32 

.45643 

2.19092 

.47769 

2.09341 

.49931 

2.00277 

.52131 

.91826 

28 

33 

.45678 

2.18923 

.47805 

2.09184 

.49967 

2.00131 

.52168 

.91690 

27 

34 

.45713 

2.18755 

.47840 

2.09028 

.50004 

1.99986 

.52205 

.91554 

26 

35 

.45748 

2.18587 

.47876 

2.08872 

.50040 

1.99841 

.52242 

.91418 

25 

36 

.45784 

2.18419 

.47912 

2.08716 

.50076 

.99695 

.52279 

.91282 

24 

37 

.45819 

2.18251 

.47948 

2.08560 

.50113 

.99550 

.52316 

.91147 

23 

38 

.45854 

2.18084 

.47984 

2.08405 

.50149 

.99406 

.52353 

.91012 

22 

39 

.45889 

2.17916 

.48019 

2.08250 

.50185 

.99261 

.52390 

.90876 

21 

40 

.45924 

2.17749 

.48055 

2.08094 

.50222 

.99116 

.52427 

.90741 

20 

41 

.45960 

2.17582 

.48091 

2.07939 

.50258 

.98972 

.52464 

1.90607 

19 

42 

.45995 

2.17416 

.48127 

2.07785 

.50295 

.98828 

.52501 

1.90472 

18 

43 

.46030 

2.17249 

.48163 

2.07630 

.50331 

.98684 

.52538 

1.90337 

17 

44 

.46065 

2.17083 

.48198 

2.07476 

1   .50368 

.98540 

.52575 

1.90203 

16 

45 

.46101 

2.16917 

.48234 

2.07321 

.50404 

.98396 

.52613 

1.90069 

15 

46 

.46136 

2.16751 

.48270 

2.07167 

.50441 

.98253 

.52650 

1.89935 

14 

47 

.46171 

2.16585 

.48306 

2.07014 

.50477 

.98110 

.52687 

1.8G801 

13 

48 

.46206 

2.16420 

.48342 

2.06860 

.50514 

.97966 

.52724 

1.89667 

12 

49 

.46242 

2.16255 

.48378 

2.06706 

.50550 

.97823 

.52761 

i.  89533 

11 

50 

.46277 

2.16090 

.48414 

2.06553 

.50587 

.97681 

.52798 

1.89400 

10 

51 

.46312 

2.15925 

.48450 

2.06400 

.50623 

.97538 

.52836 

1.89266 

9 

52 

.46348 

2.15760 

.48486 

2.06247 

.50660 

.97395 

.52873 

1.89133 

8 

53 

.46383 

2.15596 

.48521 

2.06094 

.50696 

.97253 

.52910 

1.89000 

7 

54 

.46418 

2.15432 

.48557 

2.05942 

.50733 

.97111 

.52947 

1.88867 

6 

55 

.46454 

2.15268 

.48593 

2.05790 

.50769 

.96969 

.52985 

1.88734 

5 

56 

.46489 

2.15104 

.48629 

2.05637 

.50806 

.96827 

.53022 

1.88602 

4 

57 

.46525 

2.14940 

.48665 

2.05485 

.50843 

.96685 

.53059 

1.88469 

3 

58 

.46560 

2.14777 

.48701 

2.05333 

.50879 

.96544 

.53096 

1.88337 

2 

59 

.46595 

2.14614 

.48737 

2.05182 

.50916 

.96402 

.53134 

1.88205 

1 

60 

.46631 

2.14451 

.48773 

2.05030 

.50953 

.96261 

.53171 

1.88073 

_0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

-—  - 

6 

5° 

6 

4° 

6 

3° 

6 

2° 

So 


SURVEYING. 


TABLE  VII.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


2 

8° 

2 

9° 

3 

0° 

3 

1° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang  ; 

Tang 

Cotang 

0 

.53171 

1.88073 

.55431 

1.80405 

.57735 

.73205 

.60086 

1.66428 

60 

1 

.53208 

1.87941 

.55469 

1.80281 

.57774 

.73089 

.60126 

1.66318 

59 

2 

.53246 

1.87809 

.5.5507 

.80158 

.57813 

.72973 

.60165 

.66209 

68 

3 

.53283 

1.87677 

.55545 

.80034 

.57851 

.72857 

.60205 

.66099 

57 

4 

.53320 

1.87546 

.55583 

.79911 

.57890 

.72741 

.60245 

.65990 

56 

B 

.53358 

1.87415 

.55621 

.79788 

.57929 

.72625 

.60284 

.65881 

55 

8 

.53395 

1.87283 

.55659 

.79665 

i  .57968 

.72509 

.60324 

.65772 

54 

7 

.53432 

1.87152 

.55G97 

.79542 

.58007 

.72393 

.60364 

.65663 

53 

8 

.53470 

1.87021 

.55736 

.79419 

.58046 

.72278 

.60403 

.65554 

R8 

9 

.53507 

1.86891 

.55774 

.79296 

.58085 

.72163 

.60443 

.65445 

51 

10 

.53545 

1.86760 

.55812 

.79174 

.58124 

.72047 

.60483 

.65337 

50 

11 

.53582 

1.86630 

.55850 

.79051 

.58162 

.71932 

.60522 

.65228 

49 

12 

.53620 

1.86499 

.55888 

.78929 

.58201 

.71817 

.60562 

.65120 

48 

13 

.53657 

1.86369 

.55926 

.78807 

.58240 

.71702 

i  .60602 

.65011 

47 

14 

.53694 

1.86239 

.55964 

.78685 

.58279 

.71588 

1  .60642 

.64903 

46 

15 

.53732 

1.86109 

.56003 

.78563 

.58318 

.71473 

.60681 

.64795 

45 

10 

.53769 

1.85979 

.56041 

.78441 

.58357 

.71358 

.60721 

.64687 

44 

17 

.53807 

1.85850 

.56079 

.78319 

.58396 

.71244 

.60761 

.64579 

43 

18 

.53844 

1.85720 

.56117 

.7-8198 

.58435 

.71129 

.60801 

.64471 

42 

Ifl 

.53882 

1.85591 

.56156 

.78077 

.58474 

.71015 

.60841 

.64363 

41 

20 

.53920 

1.85462 

.56194 

.77955 

.58513 

.70901 

.60881 

.64256 

40 

21 

.53957 

1.85333 

.56232 

.77834 

.58552 

.70787 

.60921 

.64148 

39 

22 

.53995 

1.85204 

.56270 

.77713 

.58501 

.70673 

.60960 

.64041 

33 

23 

.54032 

1.85075 

.56309 

.77592 

.58631 

.70560 

.61000 

.63934 

37 

24 

.54070 

1.84946 

.56347 

:  .77471 

.58670 

.70446 

.61040 

.63826 

3tj 

2r, 

.54107 

1.84818 

.56385 

.77351 

.58709 

.70332 

.61080 

.63719 

35 

20 

.54145 

1.84689 

.56424 

.77230 

.58748 

.70219 

.61120 

.63612 

34 

27 

.54183 

1.84561 

.56462 

.77110 

.58787 

.70106 

.61160 

.63505 

33 

2* 

.54220 

1.84433 

.56501 

.76990 

.58826 

.C9992 

.61200 

.63398 

32 

0 

.54258 

1.84305 

.56609 

.76869 

.58865 

.69879 

.61240 

.63292 

31 

30 

.54296 

1.84177 

.56577 

.76749 

.58905 

.69766 

.61280 

.63185 

30 

31 

.54333 

1.84049 

.56616 

.76629 

.58944 

.69653 

.61320 

.63079 

29 

82 

.54371 

1.83922 

.56654 

.76510 

.58983 

.69541 

.61360 

.62972 

28 

33 

.54409 

1.83794 

.56693 

.76390 

.59022 

.69428 

.61400 

.62866 

on 

34 

.54446 

1.83667 

.56731 

.76271 

i  .59061 

.69316 

.61440 

.62760 

26 

35 

.54484 

1.83540 

.56769 

.76151 

.59101 

.69203 

.61480 

.62654 

25 

36 

.54522 

1.83413 

.56808 

.76032 

.59140 

.69091 

.61520 

.62548 

24 

37 

.54560 

1.83286 

.56846 

.75913 

.59179 

.68979 

.61561 

.62442 

23 

88 

.54597 

1.83159 

.56885 

75794 

.59218 

.68866 

.61601 

.62336 

22 

39 

.54635 

1.83033 

.56923 

.75675 

.59258 

.68754 

.61641 

.62230 

21 

40 

.54673 

1.83906 

.56962 

.75556 

.59297 

.68643 

.61681 

.62125 

20 

41 

54711 

1.82780 

.57000 

.75437 

.59336 

.68531 

.61721 

.62019 

19 

42 

.54748 

1.82654 

.57039 

.75319 

.59376 

.68419 

.61761 

.61914 

18 

43 

.54786 

1.82528 

.57078 

.75200 

.59415 

.68308 

.61801 

.61808 

17 

44 

.54824 

1.82402 

.57116 

.75082 

.59454 

.68196 

.61842 

.61703 

16 

45 

.54862 

1.82276 

.57155 

.74964 

.59494 

.68085 

.61882 

.61598 

15 

46 

.54900 

1.82150 

.57193 

.74846 

i  .59533 

.67974 

.61922 

.61493 

14 

47 

.54938 

1.82025 

.57232 

.74728 

.59573 

.67863 

.61962 

.61388 

13 

48 

.54975 

1.81899 

.57271 

.74610 

i  .59612 

.67752 

.62003 

.61283 

12 

49 

.55013 

1.81774 

.57309 

.74492 

i  .59651 

.67641 

.62043 

.61179 

11 

50 

.55051 

1.81649 

.57348 

.74375 

j  .59691 

.67530 

.62083 

.61074 

10 

51 

.55089 

1.81524 

.57386 

.74257 

!  .59730 

.67419 

.62124 

.60970 

9 

52 

.55127 

1.81399 

.57425 

.74140 

.59770 

.67309 

.62164 

.60865 

8 

53 

.55165 

1.81274 

.57464 

.74022 

.59809 

.67198 

.62204 

:  .60761 

7 

54 

.55203 

1.81150 

.57503 

.73905 

1  .59849 

.67088 

.62245 

.60657 

6 

55 

.55241 

1.81025 

.57541 

.73788 

.59888 

.66978 

.62285 

.60553 

5 

56 

.55279 

1.80901 

.57580 

.73671 

.59928 

.66867 

.62325 

.60449 

4 

57 

.55317 

1.80777 

.57619 

.73555 

.59967 

.66757 

.62366 

.60345 

3 

58 

.55355 

1.80653 

.57657 

1.73438 

.60007 

.66647 

.62406 

.60241 

2 

59 

.53893 

1.80529 

i  .57696 

1.73321 

.60046 

.66538 

.62446 

.60137 

1 

GO 

.55431 

1.H0405 

.57735 

1.73205 

.60086 

.66428 

.62487 

60033 

0 

; 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

7} 

6 

1° 

6 

0° 

5 

9° 

5 

8° 

j 

TABLES. 


8l 


TABLE  Vll.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


32° 

33°           1 

34°            I 

35° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.62487 

1.60033 

.64941 

1.53986 

.67451 

1.48256  i 

.70021 

1.42815 

60 

1 

.62527 

1.59930 

.64982 

1.53888 

.67493 

1.48163 

.70064 

1.42726 

59 

.62568 

1.59826 

.65024 

1.53791 

.67536 

1.48070 

.70107 

1.42638 

58 

3 

.62608 

1.59723 

.65065 

1.53693 

.67578 

1.47977 

.70151 

1.42550 

57 

4 

.62649 

1.59620 

.65106 

1.53595 

.67620 

1.47885 

.70194 

1.42462 

56 

5 

.62689 

1.59517 

.65148 

1.53497 

.67063 

1.47792 

.70238 

1.42374 

55 

3 

.62730 

1.59414 

.65189 

1.53400 

.67705 

1.47699 

.70281 

1.42286 

54 

7 

.62770 

1.59311 

.65231 

1.53302 

.67748 

1.47607 

.70325 

1.42198 

53 

8 

.62811 

1.59208 

.65272 

1.53205 

.67790 

1.47514 

.70368 

1.42110 

52 

9 

.62852 

1.59105 

.65314 

1.53107 

.67832 

1.47422 

.70412 

1.42022 

51 

10 

.62892 

1.59002 

.65355 

1.53010 

.67875 

1.47330 

.70455 

1.41934 

50 

11 

.62933 

1.58900 

.65397 

1.52913 

.67917 

1.47238 

.70499 

1.41847 

49 

2 

.62973 

1.58797 

.65438 

1.52816 

.67960 

1.47146 

.70542 

1.41759 

43 

3 

.63014 

1.58695 

.65480 

1.52719 

.68002 

1.47053 

.70586 

1.41672 

47 

4 

.63055 

1.58593 

.65521 

1.52622  ! 

.68045 

1.4G9G2 

.70629 

1.41584 

4G 

3 

.63095 

1.58490 

.65563 

1.52525 

.68088 

1.43870 

.70673 

1.41497 

*5 

6 

.63136 

1.58388 

.65604 

1.52429 

.68130 

1.4G778 

.70717 

1.41409 

41 

.63177 

1.58286 

.65646 

1.52332 

.68173 

1.40086 

.70760 

1.41322 

43 

8 

.63217 

1.58184 

.65688 

1.52235 

.68215 

1.4G595 

.70804 

1.41235 

42 

19 

63258 

1.58083 

.65729 

1.52139 

.68258 

1.40503 

.70848 

1.41148 

41 

20 

63299 

1.57981 

.65771 

1.52043 

.68301 

1.4G411 

.70891 

1.41061 

40 

21 

63340 

1.57879 

.65813 

1.51946 

.68343 

1.46320 

.70985 

1.40974 

39 

22 

.63380 

1.57778 

.65854 

1.51850 

.08386 

1.4G229 

.70979 

1.40387 

23 

23 

63421 

1.57676 

.65896 

1.51754 

.08429 

1.46137 

.71023 

1.40800 

37 

24 

63462 

1.57575 

.65938 

1.51658 

.68471 

1.46046 

.71066 

1.40714 

36 

25 

.63503 

1.57474 

.G5980 

1.515G2 

.08514 

1.45955 

.71110 

1.40C27 

?J5 

26 

.63544 

1.57372 

.66021 

1.  51,406 

.00557 

1.45864 

.71154 

1.40540 

4 

27 

.63584 

1.57271 

.66063 

1.51370 

.08000 

1.45773 

.71198 

1.40454 

33 

28 

.63625 

1.57170 

.66105 

1.51275 

.08642 

1.45682 

.71242 

1.40367 

32 

29 

.63666 

1.57069 

.66147 

1.51179  1 

.08085 

1.45592 

.71285 

1.40281 

31 

30 

.63707 

1.56969 

.66189 

1.51084 

.08728 

1.45501 

.71329 

1.40195 

20 

31 

.63748 

1.56868 

.66230 

1.50988 

.08771 

1.45410 

.71373 

1  .40109 

29 

32 

.03789 

1.56767 

.66273 

1.G0803 

.08814 

1.45320 

.71417 

1.40022 

28 

33 

.63830 

1.56667 

.66314 

1.50797 

.08857 

1.45229 

.71461 

1.80936 

27 

34 

.63871 

1.56566 

.663^6 

1.50702 

.08900 

1.45139 

.71505 

1.39350 

28 

35 

.63912 

1.56466 

.66308 

1.50007 

.08942 

1.45049 

.71549 

1.39764 

25 

16 

.63953 

1.56366 

.C6440 

1.50512 

.08985 

1.44958 

.71593 

1.39G79 

24 

>7 

.63994 

1.56265 

.66482 

1.50417 

.69028 

1.44868 

.71637 

1.39593 

23 

38 

.64035 

1.56165 

.66524 

1.50322 

.69071 

1.44778 

.71681 

1.39507 

22 

39 

.64076 

1.56065 

.665GG 

1.50228 

.69114 

1.44688 

.71725 

1.39421 

21 

40 

.64117 

1.55966 

.66608 

1.50133 

.69157 

1.44598 

.71769 

1.39336 

20 

41 

.64158 

1.55866 

.66650 

1.50038 

.69200 

1.44508 

.71813 

1.39250 

19 

42 

.64199 

1.55766 

.66692 

1.49944 

.69243 

1.44418 

.71857 

1.39165 

10 

43 

.64240 

1.55666 

.66734 

1.49849 

.69286 

1.44329 

.71901 

1.39079 

17 

44 

.64281 

1.55567 

.66776 

1.49755 

.69329 

1.44239 

.71946 

1.38994 

16 

45 

.64822 

1.55467 

.66818 

1.49661 

.69372 

1.44149 

'.  71990 

1.38909 

15 

46 

.64363 

1.55368 

.66860 

1.49566 

.69416 

1.44060 

.72034 

1.38824 

14 

47 

.64404 

1.55269 

.66902 

1.49472 

.69459 

1.43970 

.72078 

1.38738 

13 

48 

.64446 

1.55170 

.66944 

1.49378 

.69502 

1.43881 

.72122 

1.38653 

12 

49 

.64487 

1.55071 

.66986 

1.49284 

.69545 

1.43792 

.72167 

1.38568 

11 

50 

.64528 

1.54972 

.67028 

1.49190 

.69588 

1.43703 

.72211 

1.38484 

10 

5 

.64569 

1.54873 

.67071 

1.49097 

.69631 

1.43614 

.72255 

1.3&399 

9 

5 

.64610 

1.54774 

.67113 

1.49003 

.69675 

1.43525 

.72299 

1.38314 

8 

5 

.64652 

1.54675 

.67155 

1.48909 

.69718 

1.43436 

.72344 

1.38229 

7 

54 

.64699 

1.54576 

.67197 

1.48816 

.69761 

1.43347 

.72388 

1.38145 

6 

5 

.64734 

1.54478 

.67239 

1.48722 

.69804 

1.43258 

.72432 

1.38060 

I 

5 

.64775 

1.54379 

.67282 

1.48629 

.69847 

1.43169 

.72477 

1.37976 

4 

5 

.64817 

1.54281 

.67324 

1.48536 

.69891 

1.43080 

.72521 

1.37891 

3 

& 

.64858 

1.54183 

.67366 

1.48442 

.69934 

1.42992 

.72565 

1.37807 

2 

5 

.64899 

1.54085 

.67409 

1.48349 

.69977 

1.42903 

.72610 

1.37722 

1 

6 

.64941 

1.53986 

1    .67451 

1.48256 

.70021 

1.42815 

.72654 

1.37638 

C 

Cotang  |   Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

i 

57° 

56° 

55° 

54° 

82 


SUX  VE  YING. 


TABLE  VII.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


3 

6° 

3 

7° 

3 

8° 

3 

9° 

,! 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.72654 

1.37638 

.75355 

1.32704 

.78129 

.27994 

.80978 

1.23490 

60 

1 

.72699 

1.37554 

.75401 

1.32624 

.78175 

.27917 

.81027 

1.23416 

59 

2 

.72743 

1.37470 

.75447 

1.32544 

.78222 

.27841 

.81075 

1.23343 

58 

3 

.72788 

1.37386 

.75492 

1.32464 

.78269 

.27764 

.81123 

1.23270 

57 

4 

.72832 

1.37302 

.75538 

1.32384 

.78316 

.27688 

.81171 

1.23196 

56 

5 

.72877 

1.37218 

.75584 

1.32304 

.78363 

.27611 

.81220 

1.23123 

55 

6 

.72921 

1.37134 

.75629 

1.32224 

.78410 

.27535 

.81268 

1.23050 

54 

7 

.72966 

1.37050 

.75675 

1.32144 

.78457 

.27458 

.81316 

1.22977 

53 

8 

.73010 

1.36967 

.75721 

1.32064 

.78504 

.27382 

.81364 

1.22904 

52 

9 

.73055 

1.36883 

.75707 

1.31984 

.78551 

.27306 

.81413 

1.22831 

51 

10 

.73100 

1.36800 

.75812 

1.31904 

.78598 

.27230 

.81461 

1.22758 

50 

11 

.73144 

1.36716 

.75858 

1.31825 

.78645 

.27153 

.81510 

1.22685 

49 

12 

.73189 

1.36633 

.75904 

1.31745 

.78CS2 

.27077 

.81558 

1.22612 

48 

13 

.73234 

1.36549 

.75950 

1.31066 

.73739 

.27001 

.81600 

1.22539 

47 

14 

.73278 

1.36466 

.75996 

1.31586 

.78786 

.26925 

.81055 

1.22467 

46 

15 

.73323 

1.36383 

.76042 

1.31507 

.78834 

.20849 

.81703 

1.22394 

45 

16 

.73368 

1.36300 

.76088 

1.31427 

.78881 

.20774 

.81752 

1.22321 

44 

17 

.73413 

1.33217 

.76134 

1.31348 

.78928 

.20098 

.81800 

1.22249 

43 

18 

.73457 

1.30134 

.76180 

1.31209 

.7897'5 

.20022 

.81849 

1.22176 

42 

19 

.73502 

1.3C351 

.76226 

1.31190 

.79022 

.20546 

.81898 

1.22104 

41 

20 

.73547 

1.35968 

.76272 

1.31110 

.79070 

.26471 

.81946 

1.22031 

40 

21 

.73592 

1.35885 

.76318 

1.31031 

.79117 

.26395 

.81995 

1.21959 

39 

22 

.73637 

1.83802 

.70304 

1.30952 

.79104 

.20319 

.82044 

1.21886 

88 

23 

.73681 

1.35719 

.76410 

1.30373 

.79212 

.26244 

.82092 

1.21814 

37 

24 

.73726 

1.35637 

.76456 

1.30795 

.79259 

.20169 

.82141 

1.21742 

36 

25 

.73771 

1.35554 

.76502 

1.30716 

.79308 

.20093 

.82190 

1.21670 

35 

26 

.73816 

1.35472 

.76548 

1.30637 

.79354 

.20018 

.82238 

1.21598 

34 

27 

.73861 

1.35389 

.76594 

1.3C558 

.79401 

.25943 

.82287 

1.21526 

33 

28 

.73906 

1.35307 

.76640 

1.30480 

.79449 

.25867 

.82336 

1.21454 

32 

29 

.73951 

1.35224 

.70686 

1.30401 

.79496 

.25792 

.82385 

1.21382 

31 

30 

.73996 

1.35142 

.76733 

1.30323 

.79544 

.25717 

.82434 

1.21310 

30 

31 

.74041 

1.35060 

.76779 

1.30244 

.79591 

.25642 

.82483 

1.21238 

29 

32 

.74086 

1.34978 

.70825 

1.30166 

.79639 

.25567 

.82531 

1.21166 

23 

33 

.74131 

1.34896 

.70871 

1.30087 

.79636 

.25492 

.82580 

1.21094 

27 

34 

.74176 

1.34814 

.76918 

1.30009 

.79734 

.25417 

.82629 

1.21023 

26 

35 

.74221 

1.34732 

.76904 

1.29931 

.79781 

.25343 

.82G7'8 

1.20951 

25 

36 

.74267 

1.34650 

.77010 

1.29853 

.79829 

.25268 

.82727 

1.20879 

24 

37 

.74312 

1.34568 

.77057 

1.20775 

.79877 

:  .25193 

.82776 

1.20808 

23 

38 

.74357 

1.34487 

.77103 

1.29696 

.79924 

.25118 

.82825 

1.20736 

22 

39 

.74402 

1.34405 

.77149 

1.29618 

.79972 

.25044 

.82874 

1.20665 

21 

40 

.74447 

1.34323 

.77196 

1.29541 

.80020 

.24969 

.82923 

1.20593 

20 

41 

.74492 

1.34242 

.77242 

1.29463 

.80067 

.24895 

.82972 

1.20522 

19 

42 

.74538 

1.34160 

.77239 

1.29385 

.80115 

.24820 

.83022 

1.20451 

18 

43 

.74583 

1.34079 

.77335 

1.29307 

.80163 

.24746 

.83071 

1.20379 

17 

44 

.74628 

1.33998 

.77382 

1.29229 

.80211 

.24672 

.83120 

1.20308 

16 

45 

.74674 

1.33916 

.77428 

1.29152 

.80258 

.24597 

.83109 

1.20237 

15 

46 

.74719 

1.33835 

.77475 

1.29074 

.80306 

.24523 

.83218 

1.20166 

14 

47 

.74764 

1.33754 

.77521 

1.28997 

.80354 

.24449 

.83268 

1.20095 

13 

48 

.74810 

1.33673 

.77568 

1.28919 

.80402 

.24375 

.83317 

1.20024 

12 

49 

.74855 

1.33592 

.77615 

1.28842 

.80450 

.24301 

.83366 

1.19953 

11 

50 

.74900 

1.33511 

.77661 

1.28764 

.80498 

.24227 

.83415 

1.19882 

10 

51 

.74946 

1.33430 

.77708 

1.28687 

.80546 

.24153 

.83465 

1.19811 

9 

52 

.74991 

1.33349 

.77754 

1.28610 

.80594 

.24079 

.83514 

1.19740 

8 

53 

.75037 

1.33268 

.77801 

1.28533 

.80642 

.24005 

.83564 

1.19669 

7 

54 

.75082 

1.33187 

.77848 

1.28456 

.80690 

.23931 

.83613 

.19599 

6 

55 

.75128 

1.33107 

.77895 

1.28379 

.80738 

.23858 

.83662 

.19528 

5 

56 

.75173 

1.33026 

.77941 

1.28302 

.80786 

.23784 

.83712 

.19457 

4 

57 

.75219 

1.32946 

.77988 

1.28225 

.80834 

.23710 

.83761 

.19387 

3 

58 

.75264 

1.32865 

.78035 

1.28148 

.80882 

.23637 

.83811 

.19316 

2 

59 

.75310 

1.32785 

.78082 

1.28071 

.80930 

.23563 

.83860 

.19246 

1 

60 

.75355 

1.32704 

.78129 

1.27994 

.80978 

.23490 

.83910 

.19175 

0 

t 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

I 

3° 

J 

2° 

5 

1° 

5 

0° 

TABLES. 


TABLE  \\l.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


J 

4 

0° 

4 

1°           I 

4 

3° 

4 

1 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~0 

.83910 

1.19175 

.86929 

1.15037 

.90040 

.11061 

.93252 

1.07237 

60 

1 

.83960 

1.19105 

.86980 

1.14969 

.90093 

.10996 

.93306 

1.07174 

59 

2 

.84009 

1.19035 

.87031 

1.14902 

.90146 

.10931 

.93360 

1.07112 

58 

3 

.84059 

1.18964 

.87082 

1.14834 

.90199 

.10867 

.93415 

1.07049 

57 

4 

.84108 

1.18894 

.87133 

1.14767 

.90251 

.10802 

.93469 

1.06987 

5G 

B 

.84158 

1.18824 

.87184 

1.14699 

.90304 

.10737 

.93524 

1.06925 

55 

6 

.84208 

1.18754 

.87236 

1.14632 

.90357 

.10672 

.93578 

1.06862 

54 

7 

.84258 

1.18684 

.87287 

1.14565 

.90410 

.10607 

.93633 

1.06800 

53 

R 

.84307 

1.18614 

.87338 

1.14498 

.90463 

.10543 

.93688 

.06738 

52 

9 

.84357 

1.18544 

.87389 

1.14430 

.90516 

.10478 

.93742 

.06676 

51 

10 

.84407 

1.18474 

.87441 

1.14363 

.90569 

.10414 

.93797 

.06613 

50 

11 

.84457 

1.18404 

.87492 

1.14296 

.00621 

.10349 

.93852 

.06551 

49 

12 

.84507 

1.18334 

.87543 

1.14229 

.90674 

.10285 

.93906 

.06489 

43 

13 

.84556 

1.18264 

.87595 

1.14162 

.90727 

.10220 

.93961 

.06427 

47 

14 

.84606 

1.18194 

.87646 

1.14095 

.90781 

.10156 

.94016 

.06365 

40 

15 

.84656 

1.18125 

.87698 

1.14028 

.90834 

.10091 

.94071 

.06303 

45 

1G 

.84706 

1.18055 

.87749 

1.13961 

.90887 

.10027 

.94125 

.06241 

44 

17 

.84756 

1.17986 

.87801 

1.13894 

.90940 

.09963 

.94180 

.06179 

43 

18 

.84806 

1.17916 

.87852 

1.13828 

.C0993 

.09899 

.94235 

.06117 

42 

19 

.84856 

1.17846 

.87904 

1.13761 

.91046 

.09834 

.94290 

.06056 

41 

90 

.84906 

1.17777 

.87955 

1.13694 

.91099 

.09770 

.94345 

.05994 

40 

21 

.84956 

1.17708 

.88007 

1.13627 

.01153 

.09706 

.94400 

.05932 

3!) 

.85006 

1.17638 

.88059 

1.13561 

.91206 

.09642 

.94455 

.05870 

38 

23 

.85057 

1.17569 

.88110 

1.13494 

.91259 

.09578 

.94510 

.05809 

37 

24 

.85107 

1.17500 

.88162 

1.13428 

.91313 

.09514 

.94565 

.05747 

36 

25 

.85157 

1.17430 

.88214 

1.13361 

.91-366 

.09450 

.94620 

.05685 

35 

20 

.85207 

1.17361 

.88265 

1.13295 

.91419 

.09386 

.94676 

.05624 

34 

27 

.85257 

1.17292 

.88317 

1.13223 

.91473 

.09322 

.94731 

.05562 

33 

2!i 

.85308 

1.17223 

.88369 

1.13162 

.91526 

.09258 

.94786 

.05501 

32 

29 

.85358 

1.17154 

.88421 

1.13096 

.91580 

.09195 

.94841 

.05439 

31 

30 

.85408 

1.17085 

.88473 

1.13029 

.91633 

.09131 

.94896 

.05378 

30 

31 

.85458 

1.17016 

.88524 

1.12963 

.91687 

.09067 

.94952 

.05317 

29 

38 

.85509 

1.16947 

.88576 

1.12897 

.91740 

.09003 

.95007 

.05255 

28 

33 

.85559 

1.16878 

.88628 

1.12831 

.91794 

.08940 

.95062 

.05194 

27 

3-i 

.85609 

1.16809 

.88680 

1.12765 

.91847 

.08876 

.95118 

.05133 

26 

35 

.85660 

1.16741 

.88732 

1.12699 

.91901 

.08813 

.95173 

.05072 

25 

36 

.85710 

1.16672 

.88784 

1.12633 

.91955 

.08749 

.95229 

.05010 

24 

37 

.85761 

1.16603 

.88836 

1.12567 

.92008 

.08686 

.95284 

.04949 

23 

88 

.85811 

1.16535 

.88888 

1.12501 

.92062 

.08622 

.95340 

.04888 

22 

39 

.85862 

1.16466 

.88940 

1.12435 

.92116 

.08559 

.95395 

.04827 

21 

40 

.85912 

1.16398 

.88993 

1.12369 

.92170 

.08496 

.95451 

.04766 

20 

41 

.85963 

1.16329 

.89045 

1.12303 

.92224 

.08432 

.95506 

.04705 

19 

42 

.86014 

1.16261 

.89097 

1.12238 

.92277 

.08369 

.95562 

.04644 

18 

43 

.86064 

1.16192 

.89149 

1.12172 

.92331 

.08306 

.95618 

.04583 

17 

44 

.86115 

1.16124 

.89201 

1.12106 

.92385 

.08243 

.95673 

.04522 

16 

45 

.86166 

1.16056 

.89253 

1.12041 

.92439 

.08179 

.95729 

.04461 

15 

40 

.86216 

1.15987 

.89306 

1.11975 

.92493 

.08116 

.95785 

.04401 

14 

47 

.86267 

1.15919 

.89358 

1.11909 

.92547 

.08053 

.95841 

.04340 

13 

48 

.86318 

1.15851 

.89410 

1.11844 

.92601 

•07990 

.95897 

.04279 

12 

49 

.86368 

1.15783 

.89463 

1  11778 

.92655 

.07927 

.95952 

.04218 

11 

50 

.86419 

1.15715 

.89515 

1.11713 

.92709 

.07864  , 

.96008 

.04158 

10 

51 

.86470 

1.15647 

.89567 

1.11643 

.92763 

.07801 

.96064 

.04097 

9 

52 

.86521 

1.15579 

.89620 

1.11582 

.92817 

.07738 

.96120 

.04036 

8 

53 

.86572 

1.15511 

.89672 

1.11517 

.92872 

.07676 

.96176 

.03976 

7 

54 

.86623 

1.15443 

.89725 

1.11452 

.92926 

.07613 

.96232 

.03915 

6 

55 

.86674 

1.15375 

.89777 

1.11387 

.92980 

.07550 

.96288 

1.03855 

5 

56 

.86725 

1.15308 

.89830 

1.11321 

.93034 

.07487  i 

.96344 

1.03794 

4 

57 

.86776 

1.15240 

.89883 

1.11256 

.93088 

.07425 

.96400 

1.03734 

3 

58 

.86827 

1.15172 

.89935 

1.11191 

.93143 

.07362 

.96457 

1.03674 

2 

59 

.86878 

1.15104 

.89988 

1.11126 

.93197 

.07299 

.96513 

1.03613 

1 

80 

.86929 

1.15037 

.90040 

1.11061 

.93252 

1.07237 

.96569 

1.03553 

J) 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

f 

i 

!   * 

9« 

4 

8° 

4 

70 

4 

rf° 

84 


SURVEYING. 


TABLE  VII.— Continued. 
NATURAL  TANGENTS  AND  COTANGENTS. 


/ 

4 

4° 

, 

i, 

4 

£° 

i 

4 

4° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

.96569 

1.03553 

60 

20 

.97700 

1.02355 

40 

40 

.98843 

1.01170 

20 

1 

2 

.966-25 
.98681 

1.03493 
1.03433 

59 
58 

21 
22 

.97756 
.97813 

1.02295 
1.02236 

89 

MS 

(41 

42 

.98901 
.98958 

1.01112 
1  01053 

19 

18 

3 

.96788 

1.03372 

57 

23 

.97870 

1.02176 

37 

143 

.99016 

1.00994 

17 

4 

.90794 

1.03312 

56 

24 

.97927 

1.02117 

86 

44 

.99073 

1.00935 

16 

5 

.9ea->o 

1.03252 

55 

25 

.97984 

.02057 

35 

45 

.99131 

1.00876 

15 

6 

.96907 

1.03192 

54 

26 

.98041 

1.01998 

34 

!46 

.99189 

1.00818 

14 

7 

.96963 

1.03132 

53 

27 

.98098 

.01939 

33 

^47 

.99247 

1.00759 

13 

8 

.97020 

1.03072 

52 

28 

.98155 

.01879 

32 

!48 

.99304 

1.00701 

12 

9 

.97076 

1.03012 

51 

29 

.98213 

.01820 

31 

149 

.99362 

1.00642 

11 

10 

.97133 

1.02952 

50 

30 

.98270 

.01761 

30 

i50 

.99420 

1.00583 

10 

11 

.97189 

1.02892 

49 

31 

.98327 

.01702 

20 

|51 

.99478 

1.00525 

9 

12 

.97246 

1.02832 

48 

32 

.98384 

.01642 

28 

153 

.99536 

1.00467 

8 

13 

.97302 

1.02772 

47 

33 

.98441 

.01583 

27 

:53 

.99594 

1.00408 

7 

14 

.97359 

1.02713 

46 

34 

.98499 

.01524 

26 

54 

.99652 

1.00350 

6 

15 

.97416 

1.02653 

45 

35 

.98556 

.01465 

25 

55 

.99710 

1.00291 

5 

16 

.97472 

1.02593 

44 

36 

.98613 

.01406 

24 

56 

.99768 

1.00233 

4 

1? 

.97529 

1.02533 

43 

37 

.98671 

.01347 

23 

57 

.99826 

1.00175 

3 

18 

.97586 

1.02474 

42 

38 

.98728 

.01288 

22 

58 

.99884 

1.00116 

2 

19 

.97643 

1.02414 

41 

39 

.98786 

.01229 

21 

59 

.99942 

1.00058 

1 

20 

.97700 

1.02355 

40 

40 

.98843 

.01170 

20 

60 

1.00000 

1.00000 

0 

Cotang 

Tang 

/ 

/ 

Cotang 

Tang 

/ 

/ 

Cotang 

Tang 

/ 

-1 

t>° 

4 

0° 

4 

5° 

- 

} 

TABLES. 


r 

i 

4^         4*        4^        4* 
OO        ON       4^-         to 
O          O          O          O 

4^       Co       Co       Co       Co       ?>J 

I 

$ 

a 

fON      CN      ON 
vo       vo       vo 
O         O         O 

vb     vb     vb      vb      oo     bo 

00       ON      4>-         1-1       vo        -J 
vj         4^          M          00        ^J         Ln 

III 

Co 

i 

Ln          OO         t-n         4^ 
^J        CO          M          O 
Ln        CO          O          OO 

VQ         VO           *"•           *^           CO        VO 

'ilH 

oo 

Co 

VO 
00 

00         OO         00        VO 
ON       ^J         ^J        Ln 

O        O        O       vo 

VO       vO       vO         O         O         O 
Co        ON       OO       I-H       Co       Ln 
Co          O         Ln          O        CO        Ln 
"M        4^.        VO          M          to         O 

1 

g; 

I 

^      ^J       oo      oo 

4*        ^J         O         M 

I-H        Ln        VO        4*. 
Ln          to         ^1          O 

OO        CC      VO        VO        VO        VO 
Ln         "^           O           tO          4^           CN 
Co          00        >-i        Co        4*.        4^. 
OO         to         Ln         *^J           OO        ^-J 

Co         to         10       Co         M        ON 

H 

Cn 

4k 

4*.        4^        4v        Ln 

ON       OO      VO         "i 

CO           IH           OO        ji. 
•x»        (.0        Co         00 

b     un      b      •£>••     vj     vb 

CN        ^J           tO           O           1-1         Ln 

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a      a      a      a 

«      s      a      a      a      a 

§ 

i 

£      S      g      S 

o       o       o       o       o       o 

00             (/i             C/3             C/3             C/3             C/5 

^ 

P 

~n 

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poop 

Lo          M         O          00 
Ln          OO         to          O\ 

o      p       p       p       p       p 

Co       Co       Co     .  Co       Co         to 
ON       Ln        CO          M          O          OO 
VO        Co         *^J          O         4^-          OO 

O           O           O           O           O           O 

0                              ™ 
0 

Ln 

Ln        Ln        Ln        Ln 

Ln        Ln        Ln         -P*-         4^         4^ 
L.J         CN      Ln       Co       ^J        O 

bo     b      -^      b      bo     4*. 

5*             0 

PI 
^               § 

to 

OO         CO         CO         00 

4».        ON      ON     Co 

b      M      M      bo 

^         ^J          CN       Ln        Co          to 

b      bo     4*.      *<i      bo     b 

?         ? 

P 
00 

poop 

p      p      p      p      p      p 

to        to        to        to        to        to 

VO         VO           OO         00        ^J           ON 
ON        M        Co         O        Cn        *>I 

8 

S              i 

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N         *"M           »          m          ta        *» 

i     i 

86 


SURVEYING. 


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«§^i&,^    5S»^RR    «g«!?gc^&    SSi^g^    S  £>££<§ 

* 

C 

J 

HNW^IO    er^wase    «^®«o    «^co»o    IH»N?OO 

•;. 

B 

^      HHHHOt      W«N«CO       CO  CC  «*  ^<  »0 

TABLES. 


j     C 

si 


g      S 

2 


z    g 

OS 


&* 

ro       Q 


fcui 

J-< 
3  " 


.H  o 


>eo 
>©o 

>00 


o       ^      «  rs.w 


O  M  «  u-ioo  o  M 


T^  °  T1^  *  ^*  Ci  M  *»  »*  ©  us  O  »ft  ft 


-  q  M  -«-vq  oo  «  vq  ON  moo  M*»H 


,  o>  cooo  »0  N  06  ^  ft  ^( 


H«w^«io©>ftOU5©»a©»o©o©©«©ia©«so©« 


88 


SURVEYING. 


TABLE   XI. 
VOLUMES  BY  THE  PRISMOID/L  FORMULA.     §  320. 


1 

HEIGHTS. 

Corrections 

1 

1 

2 

3 

4 

5 

6    7 

8 

9 

10 

for  tenths 
in  height. 

1 

0 

1 

1 

1 

2 

2 

2 

2 

3 

3 

.1 

0 

2 

1 

1 

2 

o 

3 

3 

4 

5 

6 

6 

.2 

o 

3 

1 

2 

3 

4 

5 

6 

6 

7 

8 

9 

.  3 

o 

4 

1 

2 

4 

5 

6 

7 

9 

10 

11 

12 

•4 

1 

5 

—2 

—3 

—5 

—6 

-8 

—9 

—11 

—12 

—14 

—15 

1 

6 

2 

4 

6 

7 

9 

11 

13 

15 

17 

19 

'g 

1 

7 

2 

4 

6 

9 

11 

13 

15 

17 

19 

22 

.7 

1 

8 

2 

5 

7 

10 

12 

15 

17 

20 

22 

25 

.8 

1 

9 

3 

6 

8 

11 

14 

17 

19 

22 

25 

28 

•9 

1 

10 

3 

6 

9 

12 

15 

19 

22 

25 

28 

31 

11 

3 

n 

10 

14 

17 

20 

24 

27 

31 

34 

.  i 

o 

12 

4 

7 

11 

15 

19 

22 

26 

30 

33 

37 

.2 

1 

13 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

.3 

1 

14 

4 

9 

13 

17 

22 

26 

30 

35 

39 

43 

•4 

o 

15 

—5 

—9 

—14 

—19 

—23 

—28 

—32 

—37 

—42 

—46 

.5 

2 

10 

5 

10 

15 

20 

25 

30 

35 

40 

44 

49 

.6 

3 

17 

5 

10 

16 

21 

26 

31 

37 

42 

47 

52 

.7 

3 

18 

6 

11 

17 

22 

28 

33 

39 

44 

50 

56 

.8 

4 

19 

6 

12 

18 

23 

29 

35 

41 

47 

53 

59 

•9  !   4 

20 

6 

12 

19 

25 

31 

37 

43 

49 

56 

62 

21 

6 

11 

19 

26 

32 

39 

45 

52 

58 

05 

.  T 

1 

22 

7 

It 

20 

27 

34 

41 

48 

54 

61 

68 

.  "2 

2 

23 

7 

14 

21 

28 

86 

43 

to 

57 

64 

71 

2 

24 

7 

15 

22 

30 

37 

44 

52 

59 

67 

74 

•4 

3 

25 

-8 

—15 

—23 

—31 

—39 

—46 

—54 

—62 

—69 

—77 

4 

26 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

'5 

5 

27 

8 

17 

25 

33 

42 

no 

58 

67 

75 

83 

.7 

5 

28 

9 

17 

26 

35 

43 

52 

60 

69 

78 

86 

.8 

6 

29 

9 

18 

27 

3fi 

45 

54 

63 

72 

81 

90 

•9 

7 

80 

9 

19 

28 

37 

46 

56 

65 

74 

83 

93 

31 

10 

19 

29 

38 

48 

57 

67 

77 

86 

96 

,i 

1 

32 

10 

20 

30 

40 

49 

59 

69 

79 

89 

99 

.2 

2 

33 

10 

20 

31 

41 

51 

61 

71 

81 

92 

102 

.3 

3 

34 

10 

21 

31 

42 

52 

63 

73 

84 

94 

105 

•4 

4 

35 

—11 

oo 

-32 

—43 

—54 

—65 

—76 

—86 

-97 

-108 

5 

36 

n 

22 

33 

44 

56 

67 

78 

P9 

100 

111 

g 

6 

37 

11 

23 

31 

46 

57 

69 

80 

91 

103 

114 

.7 

8 

38 

12 

23 

35 

47 

59 

70 

82 

94 

KM 

117 

.8 

9 

39 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

•9 

10 

40 

12 

25 

37 

49 

62 

74 

86 

99 

111 

123 

41 

13 

25 

28 

51 

63 

76 

89 

101 

114 

127 

.  i 

1 

42 

13 

26 

39 

M 

(55 

78 

91 

104 

117 

130 

.2 

3 

43 

13 

27 

40 

M 

66 

80 

.  93 

106 

119 

133 

•  3 

4 

44 

14 

27 

41 

54 

68 

81 

95 

109 

J  vX. 

136 

•4 

6 

45 

—14 

—28 

—42 

—56 

—69 

—83 

-97 

-111 

—125 

—139 

.5 

46 

14 

28 

43 

57 

71 

85 

99 

114 

128 

142 

.6 

8 

47 

15 

29 

44 

58 

73 

87 

102 

116 

13! 

145 

.7 

10 

48 

15 

30 

44 

59 

74 

89 

104 

119 

133 

148 

.8 

11 

49 

15 

30 

45 

60 

76 

91 

106 

121 

136 

151 

.g 

13 

50 

15 

31 

46 

62 

77 

93 

108 

123 

139 

154 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

.1 

.2 

•3 

•4 

•  5 

.6 

•  7 

.8 

•9 

Corrections  for 

0 

0 

0 

1 

1 

1 

1 

1 

1 

tenths  in  width. 

TABLES. 


89 


TABLE   XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


- 

HEIGHTS. 

Corrections 

£ 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

for  tenths 
in  height. 

51 

16 

31 

47 

63 

79 

94 

110 

126 

142 

157 

.1 

2 

52 

16 

32 

48 

64 

80 

96 

112 

128 

144 

160 

.2 

3 

53 

16 

33 

49 

65 

82 

98 

115 

131 

147 

163 

.3 

5 

54 

17 

33 

50 

67 

83 

100 

117 

133 

150 

167 

•4 

7 

55 

—17 

-34 

—51 

—68 

—85 

—102 

—119 

—136 

—153 

—170 

.5 

8 

56 

17 

35 

52 

69 

86 

104 

121 

138 

156 

173 

.6 

10 

57 

18 

35 

53 

70 

88 

106 

123 

141 

158 

•176 

.7 

12 

58 

18 

36 

54 

72 

90 

107 

125 

143 

161 

179 

.8 

14 

59 

18 

36 

55 

73 

91 

109 

127 

146 

164 

182 

•9 

15 

60 

19 

37 

56 

74 

93 

111 

130 

148 

167 

185 

61 

19 

38 

56 

75 

94 

113 

132 

151 

169 

188 

.1 

2 

62 

19 

38 

57 

77 

96 

115 

134 

153 

172 

191 

.2 

4 

63 

19 

39 

58 

78 

97 

117 

136 

156 

175 

194 

.3 

6 

64 

20 

40 

59 

79 

99 

119 

138 

158 

178 

197 

•4 

8 

65 

—20 

—40 

—60 

—80 

-100 

-120 

—140 

—160 

—181 

-201 

.5 

10 

66 

20 

41 

61 

81 

102 

122 

143 

163 

183 

204 

.6 

12 

67 

21 

41 

62 

83 

103 

124 

145 

165 

186 

207 

.7 

14 

68 

21 

42 

63 

84 

105 

126 

147 

168 

189 

210 

.8 

16 

69 

21 

43 

64 

85 

106 

128 

149 

170 

192 

213 

•9 

18 

70 

22 

43 

65 

86 

108 

130 

151 

173 

194 

216 

71 

22 

44 

66 

88 

100 

131 

153 

175 

197 

219 

.1 

2 

72 

22 

44 

67 

89 

111 

133 

156 

178 

200 

222 

.2 

5 

73 

23 

45 

68 

90 

113 

135 

158 

180 

203 

225 

.3 

7 

74 

23 

46 

69 

91 

114 

137 

160 

183 

206 

228 

•4 

9 

75 

—23 

—46 

—69 

—93 

—116 

—139 

—162 

-185 

-208 

—231 

.5 

12 

76 

23 

47 

70 

94 

117 

141 

164 

188 

211 

235 

.6 

14 

77 

24 

48 

71 

95 

119 

143 

166 

190 

214 

238 

.7 

16 

78 

24 

48 

72 

96 

120 

144 

169 

193 

217 

241 

.8 

19 

79 

24 

49 

73 

98 

122 

146 

171 

195 

219 

244 

•9 

21 

80 

25 

49 

74 

99 

123 

148 

173 

198 

222 

247 

81 

25 

50 

75 

100 

125 

150 

175 

200 

225 

250 

.1 

3 

82 

25 

51 

76 

101 

127 

152 

177 

202 

228 

253 

.2 

5 

83 

26 

51 

77 

102 

128 

154 

179 

205 

231 

256 

.3 

8 

84 

26 

52 

78 

104 

130 

156 

181 

207 

233 

259 

.4 

10 

85 

—26 

—52 

—79 

—105 

—131 

—157 

-184 

—210 

—236 

—262 

13 

86 

27 

53 

80 

106 

133 

159 

186 

212 

239 

265 

!6 

16 

87 

27 

54 

81 

107 

134 

161 

188 

215 

242 

269 

.7 

18 

88 

27 

54 

81 

109 

136 

163 

190 

217 

244 

272 

.8 

21 

89 

27 

55 

82 

110 

137 

165 

192 

220 

247 

275 

•9 

24 

90 

28 

56 

83 

111 

139 

167 

194 

222 

250 

278 

91 

28 

56 

84 

112 

140 

169 

197 

225 

253 

281 

T 

3 

92 

28 

57 

85 

114 

142 

170 

199 

227 

256 

284 

.2 

6 

93 

29 

57 

86 

115 

144 

172 

201 

230 

258 

287 

9 

94 

29 

58 

87 

116 

145 

174 

203 

232 

261 

290 

.4 

12 

95 

-29 

—59 

-88 

—117 

-147 

—176 

—205 

—235 

—264 

—293 

15 

96 

30 

59 

89 

119 

148 

178 

207 

237 

267 

296 

.6 

18 

97 

30 

60 

90 

120 

150 

180 

210 

240 

269 

299 

.7 

21 

98 

30 

60 

91 

121 

151 

181 

212 

242 

272 

302 

.8 

23 

99 

31 

61 

92 

122 

153 

183 

214 

244 

275 

306 

•9 

26 

100 

31 

62 

93 

123 

154 

185 

216 

247 

278 

309 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1O 

.1 

.2 

•  3 

•4 

•5 

.6 

•  7 

.8 

•9 

0 

0 

0 

1 

1 

1 

1 

1 

1 

Corrections  for 
tenths  in  width. 

SURVEYING. 


TABLE   XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


1 

HEIGHTS. 

Corrections 

•o 

tor  tenths 

i 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

in  height. 

l 

3 

4 

4 

4 

5 

5 

5 

6 

6 

6 

.1 

0 

2 

7 

7 

8 

9 

9 

10 

10 

11 

12 

12 

.2 

0 

3 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

.3 

0 

4 

14 

15 

16 

17 

19 

20 

21 

22 

23 

25 

.4 

1 

5 

—17 

—19 

—20 

—22 

—23 

—25 

—26 

—28 

—29 

-31 

.5 

1 

6 

20 

22 

24 

26 

28 

30 

31 

33 

35 

37 

.6 

1 

7 

24 

26 

28 

30 

32 

35 

37 

39 

41 

43 

.7 

1 

8 

27 

30 

32 

35 

37 

40 

42 

44 

47 

49 

.8 

1 

9 

31 

33 

36 

39 

42 

44 

47 

50 

53 

56 

•9 

1 

10 

34 

37 

40 

43 

46 

49 

52 

56 

59 

62 

11 

37 

41 

44 

48 

51 

54 

58 

61 

65 

68 

.i 

0 

12 

41 

44 

48 

52 

56 

59 

63 

67 

70 

74 

.2 

1 

13 

44 

48 

52 

56 

60 

64 

68 

72 

76 

80 

.3 

1 

14 

48 

52 

56 

60 

65 

69 

73 

78 

82 

86 

•4 

2 

15 

—51 

—56 

—60 

—65 

—69 

—79 

-83 

—88 

—93 

2 

16 

54 

59 

64 

69 

74 

~79 

84 

89 

94 

99 

g 

3 

17 

58 

63 

68 

73 

79 

84 

89 

94 

100 

105 

.7 

3 

18 

61 

67 

72 

78 

83 

89 

94 

100 

106 

111 

.8 

4 

19 

65 

70 

76 

82 

88 

94 

100 

106 

111 

117 

•9 

4 

20 

68 

74 

80 

86 

93 

99 

105 

111 

117 

123 

21 

71 

78 

84 

91 

97 

104 

110 

117 

123 

130 

.1 

1 

22 

75 

81 

88 

95 

102 

109 

115 

122 

129 

136 

.2 

2 

23 

78 

85 

92 

99 

106 

114 

121 

128 

135 

142 

.3 

2 

24 

81 

89 

96 

104 

111 

119 

126 

133 

141 

148 

•4 

3 

25 

—85 

-93 

—100 

—108 

—116 

-123 

—131 

—139 

—147 

—154 

.5 

4 

26 

88 

96 

104 

112 

120 

128 

136 

144 

152 

160 

.6 

5 

27 

92 

100 

108 

117 

125 

133 

142 

150 

158 

167 

.7 

5 

28 

95 

104 

112 

121 

130 

138 

147 

156 

164 

173 

.8 

6 

29 

98 

107 

116 

125 

134 

143 

152 

161 

170 

179 

•9 

7 

30 

102 

111 

120 

130 

139 

148 

157 

167 

176 

185 

31 

105 

115 

124 

134 

144 

153 

163 

172 

182 

191 

.1 

1 

82 

109 

119 

128 

138 

148 

158 

168 

178 

188 

198 

.2 

2 

33 

112 

122 

132 

143 

153 

163 

173 

183 

194 

204 

•  3 

3 

34 

115 

126 

136 

147 

157 

168 

178 

189 

199 

210 

•4 

4 

35 

—119 

-130 

—140 

—151 

—162 

—173 

—184 

—194 

-205 

-216 

.5 

5 

36 

122 

133 

144 

156 

167 

178 

189 

200 

211 

222 

.6 

6 

37 

126 

137 

148 

160 

171 

183 

194 

206 

217 

228 

.7 

8 

38 

129 

141 

152 

164 

176 

188 

199 

211 

223 

235 

.8 

9 

39 

132 

144 

156 

169 

181 

193 

205 

217 

229 

241 

•9 

10 

40 

136 

148 

160 

173 

185 

198 

210 

222 

235 

247 

41 

139 

152 

165 

177 

190 

202 

215 

228 

240 

253 

,i 

1 

42 

143 

156 

169 

181 

194 

207 

220 

233 

246 

259 

.2 

3 

43 

146 

159 

173 

186 

199 

212 

226 

239 

252 

265 

•  3 

4 

44 

149 

163 

177 

190 

204 

217 

231 

244 

258 

272 

.4 

6 

45 

—153 

-167 

-181 

-194 

—208 

—222 

—236 

—250 

—264 

—278 

7 

46 

156 

170 

185 

199 

213 

227 

241 

256 

270 

284 

.6 

8 

47 

160 

174 

189 

203 

218 

232 

247 

261 

276 

290 

'7 

10 

48 

163 

178 

193 

207 

222 

237 

252 

267 

281 

296 

.8 

11 

49 

166 

181 

197 

212 

227 

242 

257 

272 

287 

302 

•9 

13 

60 

170 

185 

201 

216 

231 

247 

262 

278 

293 

309 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

.1 

.2 

-3   |   -4 

•5 

.6 

•  7 

.8 

•9 

Corrections  for 

0 

1 

1 

a 

2 

3 

3 

4 

4 

tenths  in  width. 

TABLES. 


gi 


TABLE   XI.—  Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


3 

HEIGHTS. 

Corrections 

1 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

for  tenths 
in  height. 

51 

173 

189 

205 

220 

236 

252 

268 

283 

299 

315 

.1 

2 

52 

177 

193 

209 

225 

241 

257 

273 

289 

305 

321 

.2 

3 

53 

180 

196 

213 

229 

245 

262 

278 

294 

311 

327 

.3 

5 

54 

183 

200 

217 

233 

250 

267 

283 

300 

317 

333 

•4 

7 

55 

-187 

-204 

—221 

—238 

-255 

—272 

—289 

—306 

—323 

—340 

8 

56 

190 

207 

225 

242 

259 

277 

294 

311 

328 

346 

'g 

10 

67 

194 

211 

229 

246 

264 

281 

299 

317 

334 

352 

.7 

12 

58 

197 

215 

233 

251 

269 

286 

304 

322 

340 

358 

.8 

14 

59 

200 

219 

237 

255 

273 

291 

310 

328 

346 

364 

•9 

15 

60 

204 

222 

241 

259 

278 

296 

315 

333 

352 

370 

61 

207 

226 

245 

264 

282 

301 

320 

339 

358 

377 

.1 

2 

62 

210 

230 

249 

268 

287 

306 

325 

344 

364 

383 

.2 

4 

63 

214 

233 

253 

272 

292 

311 

331 

350 

369 

389 

.3 

6 

64 

217 

237 

257 

277 

296 

316 

336 

356 

375 

395 

•4 

8 

65 

—221 

—241 

—261 

-281 

—301 

-321 

-341 

-361 

-381 

—401 

.5 

10 

66 

224 

244 

265 

285 

306 

326 

346 

367 

387 

407 

.6 

12 

67 

227 

248 

269 

290 

310 

331 

352 

372 

393 

414 

.7 

14 

68 

231 

252 

273 

294 

315 

336 

357 

378 

399 

420 

.8 

16 

69 

234 

256 

277 

298 

319 

341 

362 

383 

405 

426 

•9 

18 

70 

238 

259 

281 

302 

324 

346 

367 

389 

410 

432 

71 

241 

263 

285 

307 

329 

351 

373 

394 

416 

438 

.1 

2 

72 

244 

267 

289 

311 

333 

356 

378 

400 

422 

444 

.2 

5 

73 

248 

270 

293 

315 

338 

360 

383 

406 

428 

451 

.3 

7 

74 

251 

274 

297 

320 

343 

365 

388 

411 

434 

457 

•4 

9 

75 

—255 

-278 

-301 

—324 

—347 

—370 

-394 

—417 

-440 

—463 

.5 

12 

76 

258 

281 

305 

328 

352 

375 

399 

422 

446 

469 

.6 

14 

77 

261 

285 

309 

333 

356 

380 

4**i 

428 

452 

475 

.7 

16 

78 

265 

289 

313 

337 

361 

385 

409 

433 

457 

481 

.8 

19 

79 

268 

293 

317 

341 

366    390 

415 

439 

463 

488 

•9 

21 

80 

272 

296 

821 

346 

370 

395 

420 

444 

469 

494 

81 

275 

300 

325 

350 

375 

400 

425 

450 

475 

500 

.1 

3 

82 

278 

304 

329 

354 

380 

405 

430 

456 

481 

506 

.2 

5 

88 

282 

307 

333 

359 

384 

410 

435 

461 

487 

512 

.3 

8 

84 

311 

337 

363 

389 

415 

441 

467 

493 

519 

.4 

10 

85 

-289 

—315 

—341 

-367 

—394 

—420 

-446 

-472 

-498 

-525 

.5 

13 

86 

292 

319 

345 

372 

398 

425 

451 

478 

504 

531 

.6 

16 

87 

295 

322 

349 

376 

403 

430 

456 

483 

510 

537 

18 

88 

299 

326 

353 

380 

407 

435 

462 

489 

516 

543 

g 

21 

89 

303 

330 

357 

385 

412 

440 

467 

494 

522 

549 

•9 

24 

90 

306 

333 

361 

389 

417 

444 

472 

500 

528 

556 

91 

309 

337 

365 

393 

421 

449 

477 

506 

•  534 

562 

.1 

3 

92 

312 

341 

369 

398 

426 

454 

483 

511 

540 

568 

.2 

6 

93 

316 

344 

373 

402 

431 

459 

488 

517 

545 

574 

.3 

9 

94 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

•4 

12 

95 

—323 

—352 

-381 

—410 

—440 

-469 

—498 

—528 

-557 

—586 

.5 

15 

96 

326 

356 

385 

415 

444 

474 

504 

533 

563 

593 

.6 

18 

97 

329 

359 

389 

419 

449 

479 

509 

539 

569 

599 

.7 

21 

98 

333 

363 

393 

423 

454 

484 

514 

544 

575 

605 

.8 

23 

99 

336 

367 

397 

428 

458 

489 

519 

550 

581 

611 

•9 

26 

100 

340 

370 

401 

432 

463 

494 

525 

556 

586 

617 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

.1 

.2 

•3 

•4 

•  5 

.6 

•  7 

.8 

•9 

Corrections  for 

0 

1 

1 

2 

2 

3 

3 

4 

4 

tenths  in  width. 

I 

92 


SURVEYING. 


TABLE    XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


1 

HEIGHTS. 

Corrections 

1 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

for  tenths 
in  height. 

2 

13 

14 

14 

15 

8 
15 

8 
16 

8 

17 

9 

17 

9 

18 

9 

19 

.1 

.2 

0 
0 

3 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

.3 

0 

4 

26 

27 

28 

30 

31 

32 

33 

35 

36 

37 

-4 

6 

—32 

—34 

-35 

-37 

—39 

-40 

—42 

—43 

—45 

—46 

.5 

6 

39 

41 

43 

44 

46 

48 

50 

52 

54 

56 

II 

7 

45 

48 

50 

52 

54 

56 

58 

60 

63 

65 

.7 

8 

52 

54 

57 

59 

62 

64 

67 

69 

72 

74 

.8 

9 

58 

61 

64 

67 

69 

72 

75 

78 

81 

83 

10 

65 

68 

71 

74 

77 

80 

83 

86 

90 

93 

11 

71 

75 

78 

81 

85 

88 

92 

95 

98 

102 

.1 

0 

12 

78 

81 

85 

89 

93 

96 

100 

104 

107 

111 

.2 

1 

13 

84 

88 

92 

96 

100 

114 

108 

112 

116 

120 

.3 

1 

14 

91 

95 

99 

104 

108 

112 

117 

121 

125 

130 

•4 

2 

15 

-97 

—102 

-106 

—111 

-116 

—120 

—125 

—130 

-134 

—139 

.5 

2 

16 

104 

109 

114 

119 

123 

128 

133 

138 

143 

148 

.6 

3 

17 

110 

115 

121 

126 

131 

136 

142 

147 

152 

157 

.7 

3 

18 

117 

122 

128 

133 

139 

144 

150 

156 

161 

167 

.8 

4 

19 

123 

129 

135 

141 

147 

152 

158 

164 

170 

176 

•9 

4 

20 

130 

136 

142 

148 

154 

160 

167 

173 

179 

185 

21 

136 

148 

149 

156 

162 

169 

175 

181 

188 

194 

.  i 

1 

22 

143 

149 

156 

163 

170 

177 

183 

190 

197 

204 

.2 

2 

23 

149 

156 

163 

170 

177 

185 

192 

199 

206 

213 

.3 

2 

24 

156 

163 

170 

178 

185 

193 

200 

207 

215 

222 

•4 

3 

25 

—162 

-170 

—177 

-185 

-193 

-201 

-208 

—216 

-224 

—231 

4 

26 

169 

177 

185 

193 

201 

209 

217 

225 

233 

241 

.6 

5 

27 

175 

183 

192 

200 

208 

217 

225 

233 

242 

250 

.7 

5 

28 

181 

190 

199 

207 

216 

225 

233 

242 

251 

259 

.8 

6 

29 

188 

197 

206 

215 

224 

233 

842 

251 

260 

269 

•9 

7 

30 

194 

204 

213 

222 

231 

241 

250 

259 

269 

278 

31 

201 

210 

220 

230 

239 

249 

258 

268 

277 

287 

, 

1 

32 

207 

217 

227 

237 

247 

257 

267 

277 

286 

296 

.2 

2 

33 

214 

224 

234 

244 

255 

265 

275 

285 

295 

306 

.3 

3 

34 

220 

231 

241 

252 

262 

273 

283 

294 

304 

315 

•4 

4 

35 

—227 

—238 

—248 

—259 

—270 

—281 

—292 

—302 

—313 

-324 

5 

36 

233 

244 

256 

267 

278 

289 

300 

311 

322 

333 

g 

6 

37 

240 

251 

263 

274 

285 

297 

308 

320 

331 

343 

.7 

8 

38 

246 

258 

270 

281 

293 

305 

317 

328 

340 

352 

.8 

9 

39 

253 

265 

277 

289 

301 

313 

325 

337 

349 

361 

•9 

10 

40 

259 

272 

284 

296 

309 

321 

333 

346 

358 

370 

41 

266 

278 

291 

304 

316 

329 

342 

354 

367 

380 

.1 

j 

42 

272 

285 

298 

311 

324 

337 

350 

363 

376 

389 

.2 

3 

43 

279 

292 

305 

319 

332 

345 

358 

372 

385 

398 

•  3 

4 

44 

285 

299 

312 

326 

340 

353 

367 

380 

394 

407 

•4 

6 

45 

—292 

—306 

—319 

—333 

—347 

—361 

—375 

—389 

-403 

—417 

.5 

7 

46 

298 

312 

327 

341 

355 

369 

383 

398 

412 

426 

.6 

8 

47 

305 

319 

334 

348 

363 

377 

392 

406 

421 

435 

.7 

10 

48 

311 

326 

341 

356 

370 

385 

400 

415 

430 

444 

.8 

11 

49 

318 

333 

348 

363 

378 

393 

408 

423 

439 

454 

•9 

13 

50 

324 

340 

355 

370 

386 

401 

417 

432 

448 

463 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

•  i 

.2 

•  3 

•4 

•  5 

.6 

•  7  ^ 

.8 

•9 

Corrections  for 

1 

2 

2 

3 

4 

5 

5 

6 

7 

tenths  in  width. 

TABLES. 


93 


TABLE    XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


± 

HEIGHTS. 

Corrections 

iM 

3 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

in  height. 

61 

331 

346 

362 

378 

394 

409 

425 

441 

456 

472 

.1 

2 

52 

337 

353 

369 

385 

401 

417 

433 

449 

465 

481 

.2 

3 

53 

344 

360 

376 

393 

409 

425 

442 

458 

474 

491 

.3 

5 

54 

350 

367 

383 

400 

417 

433 

450 

467 

483 

500 

.4 

7 

55 

-356 

-373 

—390 

—407 

-424 

—441 

-458 

—475 

—492 

—509 

•  5 

8 

56 

363 

380 

398 

415 

432 

449 

467 

484 

501 

519 

.6 

10 

57 

369 

387 

405 

422 

440 

457 

475 

493 

510 

528 

.7 

12 

58 

376 

394 

412 

430 

448 

465 

483 

501 

519 

537 

.8 

14 

59 

382 

401 

419 

437 

455 

473 

492 

510 

528 

546 

•9 

15 

60 

389 

407 

426 

444 

463 

481 

500 

519 

537 

556 

61 

395 

414 

433 

452 

471 

490 

508 

527 

546 

565 

.1 

2 

62 

402 

421 

440 

459 

478 

498 

517 

536 

555 

574 

.2 

4 

63 

408 

428 

447 

467 

486 

506 

525 

544 

564 

583 

•  3 

6 

64 

415 

435 

454 

474 

494 

514 

533 

553 

573 

593 

•4 

8 

65 

—421 

—441 

-461 

—481 

—502 

—522 

—542 

-562 

—582 

-602 

10 

66 

428 

448 

469 

489 

509 

530 

550 

570 

591 

611 

.6 

12 

67 

431 

455 

476 

496 

517 

538 

558 

579 

600 

620 

.7 

14 

68 

441 

462 

483 

504 

525 

546 

567 

588 

609 

630 

.8 

16 

69 

447 

469 

490 

511 

532  , 

554 

575 

596 

618 

639 

•9 

18 

70 

454 

475 

497 

519 

540 

562 

583 

605 

627 

648 

71 

460 

482 

504 

526 

548 

570 

592 

614 

635 

657 

.1 

2 

72 

467 

489 

511 

533 

556 

578 

600 

622 

644 

667 

.2 

5 

73 

473 

496 

518 

541 

563 

586 

608 

631 

653 

676 

.3 

7 

74 

480 

502 

525 

548 

571 

594 

617 

640 

662 

685 

.4 

9 

75 

—486 

-509 

—532 

—556 

—579 

-601 

-625 

—648 

—671 

—694 

•  S 

12 

76 

493 

516 

540 

563 

586 

610 

633 

657 

680 

704 

.6 

14 

77 

499 

523 

547 

570 

594 

618 

642 

665 

689 

713 

.7 

16 

78 

506 

530 

554 

578 

602 

626 

650 

674 

698 

722 

.8 

19 

79 

512 

536 

561 

585 

610 

634 

658 

683 

707 

731 

•9 

21 

80 

519 

543 

568 

593 

617 

642 

667 

691 

716 

741 

81 

525 

550 

575 

600 

625 

650 

675 

700 

725 

750 

j 

3 

82 

531 

557 

582 

607 

633 

658 

683 

709 

734 

759 

.2 

5 

83 

538 

564 

589 

615 

640 

666 

692 

717 

743 

769 

•  3 

8 

84 

544 

570 

596 

6-22 

648 

674 

700 

726 

752 

778 

•4 

10 

85 

—551 

—  577 

-603 

—630 

—656 

—682 

-708 

—735 

—761 

—787 

•  5 

13 

86 

557 

584 

610 

637 

664 

690 

717 

743 

770 

796 

.6 

16 

87 

564 

591 

618 

644 

671 

698 

725 

752 

779 

806 

•  7 

18 

88 

570 

598 

625 

652 

679 

706 

733 

760 

788 

815 

.8 

21 

89 

577 

604 

632 

659 

687 

714 

742 

769 

797 

824 

•9 

24 

90 

583 

611 

639 

667 

694 

722 

750 

777 

806 

833 

91 

590 

618 

646 

674 

702 

730 

758 

786 

815 

843 

.1 

3 

92 

596 

625 

653 

681 

710 

738 

767 

795 

823 

852 

.2 

6 

93 

603 

631 

660 

689 

718 

746 

775 

804 

832 

861 

•  3 

9 

94 

609 

638 

667 

696 

725 

754 

783 

812 

841 

870 

•  4 

12 

95 

-616 

-645 

-674 

—704 

—733 

—762 

—792 

—821 

-850 

—880 

•  5 

15 

96 

622 

652 

681 

711 

741 

770 

800 

830 

859 

889 

.6 

18 

97 

629 

659 

689 

719 

748 

778 

808 

838 

868 

898 

.7 

21 

98 

635 

665 

696 

756 

756 

786 

817 

847 

877 

907 

.8 

23 

99 

642 

672 

703 

733 

764 

794 

825 

856 

886 

917 

•9 

26 

100 

G48 

679 

710 

741 

772 

802 

833 

864 

895 

926 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

.1 

.2 

•3 

•  4 

•5 

.6 

•7 

.8 

•9 

Corrections  for 

1 

2 

2 

3 

4 

5 

5 

6 

7 

tenths  in  width. 

94 


SURVEYING. 


TABLE   XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


co 
JS 
"O 

HEIGHTS. 

Corrections 

g 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

for  tenths 
in  height. 

1 

10 

10 

10 

10 

11 

11 

11 

12 

12 

12 

.1 

0 

2 

19 

20 

20 

21 

22 

22 

23 

23 

24 

25 

.2 

0 

3 

29 

30 

31 

31 

32 

33 

34 

35 

36 

37 

.3 

o 

4 

38 

40 

41 

42 

43 

44 

46 

47 

48 

49 

,  A 

1 

6 

—48 

—49 

—51 

—52 

—54 

—56 

—  S7 

-59 

—60 

—62 

.5 

1 

6 

57 

59 

61 

63 

65 

67 

68 

70 

72 

74 

.6 

1 

7 

67 

69 

71 

73 

76 

78 

80 

82 

84 

86 

.7 

1 

8 

77 

79 

81 

84 

86 

89 

91 

94 

96 

97 

.8 

1 

9 

86 

89 

92 

94 

97 

100 

103 

106 

108 

111 

•9 

1 

10 

96 

99 

102 

105 

108 

111 

114 

117 

120 

123 

11 

105 

109 

112 

115 

119 

122 

126 

129 

132 

136 

.1 

0 

12 

115 

119 

122 

126 

130 

133 

137 

141 

144 

148 

.2 

1 

13 

124 

128 

132 

136 

140 

144 

148 

152 

156 

160 

.3 

1 

14 

134 

138 

143 

147 

151 

156 

160 

164 

169 

173 

•4 

2 

15 

—144 

—148 

—153 

—157 

-162 

—167 

—171 

-176 

—181 

—185 

.5 

2 

16 

153 

158 

163 

168 

173 

178 

183 

188 

193 

198 

.6 

3 

17 

163 

168 

173 

178 

183 

189 

194 

199 

205 

210 

.  7 

3 

18 

172 

178 

183 

189 

194 

200 

206 

211 

217 

222 

.8 

4 

19 

182 

188 

194 

199 

205 

211 

217 

223 

229 

235 

•9 

4 

20 

191 

198 

204 

210 

216 

222 

228 

235 

241 

247 

21 

201 

207 

214 

220 

227 

233 

240 

246 

253 

259 

.  i 

1 

22 

210 

217 

224 

231 

238 

244 

251 

258 

265 

272 

.2 

2 

23 

220 

227 

234 

241 

248 

256 

263 

270 

277 

284 

.3 

2 

24 

230 

237 

244 

252 

259 

267 

274 

281 

289 

296 

•  4 

3 

25 

—239 

—247 

—255 

—262 

-270 

—278 

-285 

—293 

—301 

—309 

.5 

4 

26 

249 

*57 

265 

273 

281 

289 

297 

305 

313 

321 

.6 

5 

27 

258 

267 

275 

283 

292 

300 

308 

317 

325 

333 

.7 

5 

28 

90S 

277 

285 

294 

302 

311 

320 

328 

337 

346 

.8 

6 

29 

277 

286 

295 

304 

313 

322 

331 

340 

349 

358 

•9 

30 

287 

2% 

306 

315 

324 

3.3S 

343 

352 

361 

370 

31 

297 

306 

316 

825 

335 

344 

354 

364 

373 

383 

.1 

1 

32 

306 

316 

326 

336 

346 

356 

365 

375 

385 

395 

.2 

2 

33 

316 

326 

336 

346 

356 

367 

377 

387 

397 

407 

.3 

3 

34 

325 

336 

346 

357 

367 

378 

388 

399 

409 

420 

•  4 

4 

35 

--335 

—346 

-356 

—367 

—378 

—389 

—400 

^ilO 

—421 

-432 

.5 

5 

36 

344 

356 

367 

378 

389 

400 

411 

422 

433 

444 

.6 

6 

37 

354 

365 

377 

388 

400 

411 

423 

434 

445 

457 

.7 

8 

38 

364 

375 

387 

399 

410 

422 

434 

446 

457 

469 

.8 

9 

39 

373 

385 

S97 

409 

421 

433 

445 

457 

469 

481 

•9 

10 

40 

383 

395 

407 

420 

432 

444 

457 

469 

481 

494 

41 

392 

405 

418 

430 

443 

456 

468 

481 

494 

506 

.1 

1 

42 

402 

415 

428 

441 

454 

467 

480 

493 

506 

519 

.2 

3 

43 

411 

425 

438 

451 

465 

478 

491 

504 

518 

531 

.3 

4 

44 

421 

435 

448 

462 

475 

489 

502 

516 

530 

543 

•4 

6 

45 

-^131 

—444 

-458 

—472 

—486 

—500 

—514= 

—528 

-542 

-556 

.5 

7 

46 

440 

454 

469 

483 

497 

511 

525 

540 

554 

568 

.6 

8 

47 

450 

464 

479 

493 

508 

522 

537 

551 

566 

580 

.7 

10 

48 

459 

474 

489 

504 

519 

533 

548 

563 

578 

593 

.8 

11 

49 

469 

484 

499 

514 

529 

544 

560 

575 

590 

605 

.9 

13 

60 

478 

494 

509 

525 

540 

556 

571 

586 

602 

617 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

.1 

.2 

•3 

•4 

•  5 

.6 

•7 

.8 

•9 

Corrections  for 

1 

9 

3 

4 

5 

6 

8 

9 

10 

tenths  in  width. 

TABLES. 


95 


TABLE   XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


j3 

HEIGHTS.                         Corrections 

for  tenths 

!2 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

in  height. 

51 

488 

504 

519 

535 

551 

567 

582 

598 

614 

630 

T 

2 

62 

498 

514 

530 

546 

562 

578 

594 

610 

626 

642 

.2 

3 

63 

507 

523 

540 

556 

573 

589 

605 

622 

638 

654 

.3 

5 

64 

517 

533 

550 

567 

583 

600 

617 

633 

650 

667 

•4 

7 

66 

—526 

—543 

—560 

—577 

—594 

—611 

—628 

—645 

—662 

—679 

.5 

8 

66 

536 

553 

570 

588 

605 

622 

640 

657 

674 

691 

.6 

10 

67 

545 

563 

581 

598 

616 

633 

651 

669 

686 

704 

.7 

12 

68 

555 

573 

591 

609 

627 

644 

662 

680 

698 

716 

.8 

14 

69 

565 

583 

601 

619 

637 

656 

674 

692 

710 

728 

•9 

15 

60 

574 

593 

611 

630 

648 

667 

685 

704 

722 

741 

61 

584 

602 

621 

640 

659 

678 

697 

715 

734 

753 

.z 

2 

62 

593 

612 

631 

651 

670 

689 

708 

727 

746 

765 

.2 

4 

63 

603 

622 

642 

661 

681 

700 

719 

739 

758 

778 

.3 

:   6 

64 

612 

632 

652 

672 

691 

711 

731 

751 

770 

790 

•4 

8 

66 

—622 

—642 

—662 

—682 

—702 

—722 

—742 

—762 

—782 

—802 

.5 

10 

66 

631 

652 

672 

693 

713 

733 

754 

774 

794 

815 

.6 

12 

67 

611 

662 

682 

703 

724 

744 

765 

786 

806 

827 

.7 

14 

68 

651 

672 

693 

714 

735 

756 

777 

798 

819 

840 

.8 

16 

69 

660 

681 

703 

724 

745 

767 

788 

809 

831 

852 

•9 

18 

70 

670 

691 

713 

735 

756 

778 

799 

821 

843 

864 

71 

679 

roi 

723 

745 

767 

789 

811 

833 

855 

877 

, 

2 

72 

689 

711 

733 

756 

778 

800 

822 

844 

867 

889 

.2 

5 

73 

698 

721 

744 

766 

789 

811 

834 

856 

879 

901 

•  3 

7 

74 

708 

731 

754 

777 

799 

822 

845 

868 

891 

914 

•4 

9 

75 

—718 

—741 

—764 

—787 

—810 

—833 

—856 

-€80 

—903 

—926 

.5 

12 

76 

727 

751 

774 

798 

821 

844 

868 

891 

915 

938 

.6 

14 

77 

737 

760 

784 

808 

832 

856 

879 

903 

927 

951 

.7 

16 

78 

746 

770 

794 

819 

843 

867 

891 

915 

939 

963 

.8 

19 

79 

756 

780 

805 

829 

853 

878 

902 

927 

951 

975 

•9 

21 

80 

765 

790 

815 

840 

864 

889 

914 

938 

963 

988 

81 

775 

800 

825 

850 

875 

900 

925 

950 

975 

1000 

.1 

3 

82 

785 

810 

835 

860 

886 

911 

936 

962 

987 

1012 

.2 

5 

83 

794 

820 

845 

871 

897 

922 

948 

973 

999 

1025 

.3 

8 

84 

804 

830 

856 

881 

907 

933 

959 

985 

1011 

1037 

•4 

10 

85 

—813 

—840 

—866 

—892 

—918 

—944 

—971 

—997 

—1023 

—1049 

13 

86 

823 

849 

876 

902 

929 

956 

982 

1009 

1035 

1062 

.6 

16 

87 

832 

859 

886 

913 

940 

967 

994 

1020 

1047 

1074 

.7 

18 

88 

842 

869 

896 

923 

951 

978 

1005 

1032 

1059 

1086 

.8 

21 

89 

852 

879 

906 

934 

961 

989 

1016 

1044 

1071 

1098 

•9 

24 

90 

861 

889 

917 

944 

972 

1000 

1028 

1056 

1083 

1111 

91 

871 

899 

927 

955 

983 

1011 

1039 

1067 

1095 

1123 

.1 

3 

92 

880 

909 

937 

965 

994 

1022 

1051 

1079 

1107 

1136 

.2 

6 

93 

890 

919 

947 

976 

1005 

1033 

1062 

1091 

1119 

1148 

.3 

9 

94 

899 

928 

957 

986 

1015 

1044 

1073 

1102 

1131 

1160 

•4 

12 

95 

—909 

—938 

—968 

—997 

—1026 

—1056 

—1085 

—1114 

—1144 

—1173 

.5 

15 

96 

919 

948 

978 

1007 

1037 

1067 

1096 

1126 

1156 

1185 

.6 

18 

97 

928 

958 

988 

1018 

1048 

1078 

1108 

1138 

1168 

1198 

.7 

21 

98 

933 

968 

998 

1028 

1059 

1089 

1119 

1149 

1180 

1210 

.8 

23 

99 

947 

978 

1008 

1039 

1069 

1100 

1131 

1161 

1192 

1222 

•9 

26 

100 

957 

988 

1019 

1049 

1080 

1111 

1142 

1173 

1204 

1235 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

.1 

.2 

•3 

•4 

•5 

.6 

•7 

.8 

•9 

Corrections  for 

1 

2 

3 

4 

5 

6 

8 

9 

10 

tenths  in  width. 

96 


SURVEYING. 


TABLE   XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


1 

*o 

HEIGHTS. 

Corrections 

% 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

for  tenths 
in  height. 

1 

13 

13 

13 

14 

14 

14 

15 

15 

15 

15 

.  i 

o 

2 

25 

26 

27 

27 

28 

28 

29 

30 

30 

31 

.2 

o 

3 

38 

39 

40 

41 

42 

43 

44 

44 

45 

46 

.  -3 

o 

4 

51 

52 

53 

54 

56 

57 

58 

59 

60 

62 

*J 

1 

6 

.—63 

—65 

—  C6 

—68 

—69 

—71 

—73 

—74 

—76 

—77 

"T 

1 

6 

76 

78 

80 

81 

83 

85 

87 

89 

91 

93 

°g 

1 

7 

89 

91 

93 

95 

97 

99 

102 

104 

106 

108 

.7 

1 

8 

101 

104 

106 

109 

111 

114 

116 

119 

121 

123 

.8 

1 

9 

114 

117 

119 

122 

125 

128 

131 

133 

136 

139 

.Q 

1 

10 

127 

130 

133 

136 

139 

142 

145 

148 

151 

154 

•y 

11 

139 

143 

146 

149 

153 

156 

160 

163 

166 

170 

,i 

o 

12 

152 

156 

159 

163 

167 

170 

174 

178 

181 

185 

.2 

1 

13 

165 

169 

173 

177 

181 

185 

189 

193 

197 

201 

.  o 

1 

14 

177 

181 

186 

190 

194 

199 

203 

207 

212 

216 

"  j 

•4 

2 

15 

—190 

—194 

—199 

—204 

—208 

—213 

—218 

—222 

—227 

—231 

.  c 

2 

16 

203 

207 

212 

217 

222 

227 

232 

237 

242 

247 

.6 

3 

17 

215 

220 

226 

231 

236 

241 

247 

252 

257 

262 

.7 

3 

18 

228 

23d 

239 

244 

250 

256 

261 

267 

272 

278 

.8 

4 

19 

240 

246 

252 

258 

264 

270 

276 

281 

287 

293 

•9 

4 

20 

253 

259 

265 

272 

278 

284 

290 

296 

302 

309 

21 

266 

272 

279 

285 

292 

298 

305 

311 

318 

324 

.1 

1 

22 

278 

285 

293 

299 

306 

312 

319 

326 

333 

340 

.2 

2 

23 

291 

298 

305 

312 

319 

327 

334 

341 

348 

355 

.3 

2 

24 

304 

311 

319 

326 

333 

341 

348 

356 

363 

370 

•4 

3 

25 

—316 

—324 

—332 

—340 

-347 

—355 

—363 

—370 

—378 

—386 

.5 

4 

26 

329 

337 

345 

353 

361 

369 

377 

385 

393 

401 

.6 

5 

27 

342 

330 

358 

367 

375 

383 

392 

400 

408 

417 

.7 

5 

28 

354 

363 

372 

380 

389 

398 

406 

415 

423 

432 

.8 

6 

29 

367 

376 

385 

394 

403 

412 

421 

430 

439 

448 

•9 

7 

30 

380 

389 

398 

407 

417 

426 

435 

444 

454 

463 

31 

392 

402 

411 

421 

431 

440 

450 

459 

469 

478 

.1 

1 

32 

405 

415 

425 

435 

444 

454 

464 

474 

484 

494 

.2 

2 

33 

418 

428 

438 

448 

458 

469 

479 

489 

499 

509 

.3 

3 

34 

430 

441 

451 

462 

472 

483 

493 

504 

514 

525 

•4 

4 

35 

-443 

-454 

—465 

-475 

-486 

—497 

—508 

-519 

—529 

-540 

.5 

5 

36 

456 

467 

478 

489 

500 

511 

522 

533 

544 

556 

.6 

6 

37 

468 

480 

491 

502 

514 

525 

537 

548 

560 

671 

.7  ' 

8 

38 

481 

493 

504 

516 

528 

540 

551 

563 

575 

586 

.8 

9 

39 

494 

506 

518 

530 

542 

554 

566 

578 

590 

602 

•9 

10 

40 

506 

519 

531 

543 

556 

568 

580 

593 

605 

617 

41 

519 

531 

544 

557 

569 

582 

595 

607 

620 

633 

.1 

1 

42 

531 

544 

557 

670 

583 

596 

609 

622 

635 

648 

.2 

3 

43 

544 

557 

571 

584 

597 

610 

624 

637 

650 

664 

.3 

4 

44 

557 

570 

584 

598 

611 

625 

638 

652 

665 

679 

•4 

6 

46 

—569 

—583 

-597 

—611 

—625 

—639 

—653 

-667 

-681 

—694 

.5 

7 

46 

582 

596 

610 

625 

639 

653 

667 

681 

696 

710 

.6 

8 

47 

595 

609 

624 

638 

653 

667 

682 

696 

711 

725 

.7 

10 

48 

607 

622 

637 

652 

667 

681 

696 

711 

726 

741 

.8 

11 

49 

620 

635 

650 

665 

681 

696 

710 

726 

741 

756 

•9 

13 

60 

633 

648 

C64 

679 

694 

710 

725 

741 

756 

772 

41 

42 

43" 

44 

45 

46 

47 

48 

49 

50 

.1 

.2 

•3 

•4 

•  5 

.6 

•  7 

.8 

•9 

Corrections  for 

1 

3 

4 

6 

7 

8 

10 

11 

13 

tenths  in  width. 

TABLES. 


97 


TABLE   XI.— Continued. 
VOLUMES  BY  THE  PRISMOIDAL  FORMULA. 


i 

HEIGHTS. 

Corrections 

s 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

in  height. 

51 

645 

661 

677 

693 

708 

724 

740 

756 

771 

787 

x 

2 

52 

658 

674 

690 

706 

722 

738 

754 

770 

786 

802 

.2 

8 

53 

671 

687 

703 

720 

736 

752 

768 

785 

802 

818 

.3 

5 

54 

683 

700 

717 

733 

750 

767 

783 

800 

817 

833 

•4 

7 

55 

—696 

—713 

—730 

—747 

—764 

—781 

—798 

—815 

-832 

—849 

8 

56 

709 

726 

743 

760 

778 

795 

812 

830 

847 

864 

'.6 

10 

57 

721 

739 

756 

774 

792 

809 

827 

844 

862 

880 

.7 

12 

58 

734 

752 

770 

788 

806 

823 

841 

859 

877 

895 

.8 

14 

59 

747 

765 

783 

801 

819 

833 

856 

874 

892 

910 

•9 

15 

60 

759 

778 

706 

815 

833 

852 

870 

889 

907 

926 

61 

772 

791 

810 

828 

847 

866 

885 

991 

923 

941 

.1 

2 

62 

785 

804 

823 

842 

861 

880 

899 

919 

938 

957 

.2 

4 

63 

797 

817 

836 

856 

875 

894 

914 

933 

953 

972 

.3 

6 

64 

810 

830 

849 

869 

889 

909 

928 

948 

968 

988 

•4 

8 

65 

—823 

—843 

—863 

—883 

—903 

—923 

—943 

—963 

—983 

—1003 

10 

66 

835 

856 

876 

896 

917 

937 

957 

978 

998 

1019 

.6 

12 

67 

848 

869 

889 

910 

931 

951 

972 

993 

1013 

1034 

.7 

14 

68 

860 

881 

902 

923 

944 

965 

986 

1007 

1028 

1049 

.8 

16 

69 

873 

894 

916 

937 

958 

980 

1001 

1022 

1044 

1065 

•9 

18 

70 

886 

907 

929 

951 

972 

994 

1015 

1037 

1059 

1080 

71 

898 

920 

942 

964 

986 

1008 

1030 

1052 

1074 

1096 

.1 

2 

72 

911 

933 

956 

9^8 

1000 

1022 

1044 

1067 

1089 

1111 

.2 

3 

73 

924 

946 

969 

991 

1014 

1036 

1059 

1081 

1104 

1127 

.3 

7 

74 

936 

959 

982 

1005 

1028 

1051 

1073 

1096 

1119 

1142 

.4 

9 

15 

—949 

—972 

—995 

—1019 

—1042 

—1065 

—1088 

—1111 

—1134 

—1157 

•  5 

12 

76 

962 

985 

1009 

1032 

1056 

1079 

1102 

1126 

1149 

1173 

.6 

14 

77 

974 

998 

1022 

1046 

1069 

1093 

1117 

1141 

1165 

1188 

.7 

16 

78 

987 

1011 

1035 

1059 

1083 

1107 

1131 

1156 

1180 

1204 

.8 

19 

79 

1000 

1024 

1048 

1073 

1097 

1122 

1146 

1170 

1195 

1219 

•9 

21 

80 

1012 

1037 

1062 

1086 

1111 

1136 

1160 

1185 

1210 

1235 

81 

1025 

1050 

1075 

1100 

1125 

1150 

1175 

1200 

1225 

1250 

.1 

3 

82 

1038 

1063 

1088 

1114 

1139 

1164 

1190 

1215 

1240 

1265 

.2 

5 

83 

1050 

1076 

1102 

1127 

1153 

1178 

1204 

1230 

1255 

1281 

.3 

8 

84 

1063 

1089 

1115 

1141 

1167 

1193 

1219 

1244 

1270 

1296 

•4 

10 

85 

—1076 

—1102 

—1128 

-1154 

—1181 

—1207 

—1233 

—1259 

—1285 

—1312 

.5 

13 

86 

1088 

1115 

1141 

1168 

1194 

1221 

1248 

1274 

1301 

1327 

.6 

16 

87 

1101 

1128 

1155 

1181 

1208 

1235 

1262 

1289 

1316 

1343 

.7 

18 

88 

1114 

1141 

1168 

1195 

1222 

1249 

1277 

1304 

1331 

1358 

.8 

21 

89 

1126 

1154 

1181 

1209 

1236 

1264 

1291 

1319 

1346 

1373 

•9 

24 

90 

1139 

1167 

1194 

1222 

1250 

1278 

1806 

1333 

1361 

1389 

91 

1152 

1180 

1208 

1236 

1264 

1292 

1320 

1348 

1376 

1404 

.1 

3 

92 

1164 

1193 

1221 

1249 

1278 

1306 

1335 

1363 

1391 

1420 

6 

93 

1177 

1206 

1234 

1263 

1292 

1320 

1349 

1378 

1406 

1435 

.3 

9 

94 

1190 

1219 

1248 

1277 

1306 

1335 

1364 

1393 

1422 

1451 

.4 

12 

95 

—1202 

—1231 

—1261 

—1290 

—1319 

—1349 

—1378 

—1407 

-1437 

—1466 

.5 

15 

96 

1215 

1244 

1274 

1304 

1333 

1363 

1393 

1422 

1452 

1481 

.6 

18 

97 

1227 

1257 

1287 

1317 

1347 

1377 

1407 

1437 

1467 

1497 

.7 

21 

98 

1240 

1270 

1301 

1331 

1361 

1391 

1422 

1452 

148fi 

1512 

.8 

23 

99 

1253 

1283 

1314 

1344 

1375 

1406 

1436 

1467 

1497 

1528 

•9 

26 

100 

1265 

1296 

1327 

1358 

1389 

1420 

1451 

1481 

1512 

1543 

41 

42 

43 

44 

45 

46 

41 

48 

49 

5O 

.1 

.2 

•3 

•4 

•  s 

.6 

•7 

.8 

•9 

Corrections  for 

1 

3 

4 

6 

7 

8 

10 

11 

13 

tenths  in  width. 

98 


SURVEYING. 


TABLE  XII.— AZIMUTHS  or  POLARIS 


THE  STAR  AND  THE  AZIMUTH  are  W.  of  N.  when  the  hour  angle  is  less 

THE  ARGUMENT  is  the  star's  hour  angle  (or  23h.  56min. 

To  FIND  THE  TRUE  MERIDIAN  the  azimuth  must  he  laid  off  to  the  east  when  the 


jj 

i 

0* 

0 

<H' 

1 

m. 

i 

m. 

CO 

| 

771. 

1H 

m. 

Azimuths  for  latitude— 

Date. 
1893. 

30 

0    / 

0  2 
3 
5 
6 
8 
9 
11 
12 
14 
15 
17 
19 
20 
22 
23 
25 
27 
29 
31 
32 
34 
36 
38 
39 
41 
43 
45 
46 
48 
50 
51 
53 
54 
56 
57 
0  59 
1  0 
2 
3 
5 
6 
8 
9 
11 
12 
14 
15 
17 
19 
20 
22 
24 
26 
27 
1  89 

32 

34 

o   / 
0  2 
3 
5 

y 

8 
10 
11 
13 
15 
16 
18 
19 
21 
23 
24 
26 
28 
30 
32 
34 
36 
38 
40 
41 
43 
45 
47 
49 
50 
52 
54 
55 
57 
0  58 
1  0 
2 
3 
5 
6 
8 
9 
11 
13 
14 
16 
17 
19 
21 
23 
24 
26 
28 
30 
31 
1  32 

36 

0    / 

0  2 
2 

7 
8 
10 
12 
13 
15 
17 
18 
20 
22 
23 
25 
27 
29 
31 
33 
35 
37 
39 
41 
42 
44 
46 
43 
50 
51 
53 
55 
57 
0  58 
1  0 
2 
3 
5 
6 
8 
10 
11 
13 
14 
16 
17 
19 
21 
23 
24 
26 
28 
30 
32 
33 
1  35 

38 

0  „  / 

0  2 

3 
5 
7 
9 
10 
12 
14 
15 
17 
19 
21 
22 
24 
26 
28 
30 
62 
34 
36 
38 
40 
42 
44 
46 
47 
49 
51 
53 
55 
57 
0  58 
1  0 
2 
3 
5 
7 

8 
10 
12 
13 
15 
16 
18 
20 
21 
23 
25 
27 
2& 
30 
33 
34 
36 
1  37 

40 

42 

44 

46 

o  / 

0  2 

4 
6 
8 
10 
12 
14 
16 
18 
20 
21 
23 
25 
27 
29 
31 
34 
36 
38 
41 
43 
45 
47 
50 
52 
54 
56 
0  58 
1  0 
2 
4 
6 
8 
10 
12 
14 
16 
17 
19 
21 
23 
25 
27 
29 
31 
32 
34 
36 
39 
41 
42 
45 
47 
49 
1  50 

48 

0   / 

0  2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
23 
24 
26 
28 
30 
33 
35 
38 
40 
42 
45 
47 
49 
52 
54 
56 
0  58 
1  1 
3 
5 
7 
9 
11 
13 
15 
17 
19 
20 
22 
24 
27 
28 
30 
32 
34 
36 
38 
40 
43 
45 
47 
49 
51 
53 
155 

50 

m. 

0    / 
A   0 

o   / 
0  2 
4 
5 
7 
9 
11 
12 
14 
16 
18 
19 
22 
23 
25 
26 
28 
31 
33 
35 
37 
39 
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99 


FOR  ALL  HOUK  ANGLES.    §  381  A. 


than  llh  58m  and  E.  of  N.  when  the  hour  angle  is  greater  than  lib  58*. 

minus  the  star's  hour  angle),  for  the  years  given. 

hour  angle  is  less  than  11*  58m,  and  to  the  west  when  it  is  greater  than  llh  58m. 


°ZS£a,° 

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38 

40 

42 

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5  37.0 
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3  34.6 
2  39.4 
1  44.3 
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23  38.4 
22  35.5 
21  40.6 
20  34.0 
19  39.1 
18  36.5 
17  41.6 
16  35.1 
15  40.2 
14  33.6 
13  38.7 
12  35.9 
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-8- 

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1 

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te. 

98 

25 
27 
1  29 

6 

|  6 

bl. 
26 

1 

1 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


3  ling 


MAR 


APR  8 


6 


30m-l,'15 


